Overview of Macroscopic Thermal Sciences

Abstract

This chapter provides a concise description of the basic concepts and theories underlying classical thermodynamics and heat transfer. Different approaches exist in presenting the subject of thermodynamics. Most engineering textbooks first introduce temperature, then discuss energy, work, and heat, and define entropy afterward. An overview of classical thermodynamics is provided that is somewhat beyond typical undergraduate textbooks. The basic phenomena and governing equations in energy, mass, and momentum transfer are subsequently presented in a self-consistent manner without invoking microscopic theories.

Problems

• 2.1. Give examples of steady state. Give examples of thermodynamic equilibrium state. Give an example of spontaneous process. Is the growth of a plant a spontaneous process? Give an example of adiabatic process.

• 2.2. What is work? Describe an experiment that can measure the amount of work. What is heat? Describe an apparatus that can be used to measure heat. Are work and heat properties of a system?

• 2.3. Expand Eqs. (2.1a, 2.1b) and (2.2a) in terms of the rate of energy and entropy change of an open system, which is subjected to work output, heat interactions, and multiple inlets and outlets of steady flow.

• 2.4. Discuss the remarks of Rudolf Clausius in 1867:

  • (a) The energy of the universe is constant.

  • (b) The entropy of the universe strives to attain a maximum value.

• 2.5. For a cyclic device experiencing heat interactions with reservoirs at $T_{1} ,T_{2} ,\; \ldots$, the Clausius inequality can be expressed as $\sum\nolimits_{i} {\frac{{\delta Q_{i} }}{{T_{i} }}\; \le \;0}$ or $\oint {\frac{\delta Q}{T}} \le 0$, regardless of whether the device produces or consumes work. Note that $\delta Q$ is positive when heat is received by the device. Prove the Clausius inequality by applying the second law to a closed system.

• 2.6. In the stable-equilibrium states, the energy and the entropy of a solid are related by $E = 3 \times 10^{5} \exp \left( {\frac{{S - S_{0} }}{1000}} \right)$, where E is in J, S is in J/K, and S₀ is the entropy of the solid at a reference temperature of 300 K. Plot this relation in an E–S graph. Find expressions for E and S in terms of its temperature T and S0.

• 2.7. For an isolated system, give the mathematical expressions of the first and second laws of thermodynamics. Give graphic illustrations using E–S graph.

• 2.8. Place two identical metal blocks A and B, initially at different temperatures, in contact with each other but without interactions with any other systems. Assume thermal equilibrium is reached quickly and let system C represents the combined system of both A and B.

  • (a) Is the process reversible or not? Which system has experienced a spontaneous change of state? Which systems have experienced an induced change of state?

  • (b) Assume that the specific heat of the metal is independent of temperature, $c_{p} =$ 240 J/kg K, the initial temperatures are $T_{{{\text{A}}1}} = 800$ K and $T_{{{\text{B}}1}} = 200$ K, and the mass of each block is 5 kg. What is the final temperature? What is the total entropy generation in this process?

  • (c) Show the initial and final states of systems A, B, and C in a u–s diagram, and indicate which state is not an equilibrium state. Determine the adiabatic availability of system C in the initial state.

• 2.9. Two blocks made of the same material with the same mass are allowed to interact with each other but isolated from the surroundings. Initially, block A is at 800 K and block B at 200 K. Assuming that the specific heat is independent of temperature, show that the final equilibrium temperature is 500 K. Determine the maximum and minimum entropies that may be transferred from block A to block B.

• 2.10. A cyclic machine receives 325 kJ heat from a 1000 K reservoir and rejects 125 kJ heat to a 400 K reservoir in a cycle that produces 200 kJ work. Is this cycle reversible, irreversible, or impossible?

• 2.11. ${\text{If }}z = z(x,y),{\text{ then d}}z = f{\text{d}}x + g{\text{d}}y,{\text{ where }}f(x,y) = \partial z/\partial x, \, g(x,y) = \partial z/\partial y.$ ${\text{Therefore, }}\frac{\partial f}{\partial y} = \frac{{\partial^{2} z}}{\partial y\partial x} = \frac{{\partial^{2} z}}{\partial x\partial y} = \frac{\partial g}{\partial x} \, .$ The second-order derivatives of the fundamental equation and each of the characteristic function yield a Maxwell relation. Maxwell’s relations are very useful for evaluating the properties of a system in the stable-equilibrium states. For a closed system without chemical reactions, we have ${\text{d}}N_{i} \equiv 0.$ Show that $\left( {\frac{\partial T}{\partial V}} \right)_{S} = - \left( {\frac{\partial P}{\partial S}} \right)_{V}$, $\left( {\frac{\partial T}{\partial P}} \right)_{S} = \left( {\frac{\partial V}{\partial S}} \right)_{P} , \, \left( {\frac{\partial S}{\partial V}} \right)_{T} = \left( {\frac{\partial P}{\partial T}} \right)_{V} ,{\text{ and }}\left( {\frac{\partial S}{\partial P}} \right)_{T} = - \left( {\frac{\partial V}{\partial T}} \right)_{P}$.

• 2.12. The isobaric volume expansion coefficient is defined as $\beta_{P} = \frac{1}{v}\left( {\frac{\partial v}{\partial T}} \right)_{P}$, the isothermal compressibility is $\kappa_{T} = - \frac{1}{v}\left( {\frac{\partial v}{\partial P}} \right)_{T}$, and the speed of sound is $v_{\text{a}} = \sqrt {\left( {\frac{\partial P}{\partial \rho }} \right)_{s} }$. For an ideal gas, show that $\beta_{P} = 1/T$, $\kappa_{T} = 1/P$, and $v_{\text{a}} = \sqrt {\gamma RT}$.

• 2.13. For a system with single type of constituents, the fundamental relation obtained by experiments gives $S = \alpha (NVU)^{1/3}$, where α is a positive constant, and N, V, S, and U are the number of molecules, the volume, the entropy, and the internal energy of the system, respectively. Obtain expressions of the temperature and the pressure in terms of N, V, U, and α. Show that S = 0 at zero temperature for constant N and V.

• 2.14. For blackbody radiation in an evacuated enclosure of uniform wall temperature T, the energy density can be expressed as $u_{v} = \frac{U}{V} = \frac{4}{c}\sigma_{\text{SB}} T^{4}$, where U is the internal energy, V the volume, $c$ the speed of light, and $\sigma_{\text{SB}}$ the Stefan–Boltzmann constant. Determine the entropy $S(T, \, V)$ and the pressure $P(T, \, V)$, which is called the radiation pressure. Show that the radiation pressure is a function of temperature only and negligibly small at moderate temperatures. Hint: $S = \int_{0}^{T} {\frac{1}{T}} \left( {\frac{\partial U}{\partial T}} \right)_{V} \;{\text{d}}T$ and $P = T\left( {\frac{\partial S}{\partial V}} \right)_{T} - \left( {\frac{\partial U}{\partial V}} \right)_{T}$.

• 2.15. A cyclic machine can only interact with two reservoirs at temperatures T_A = 298 K and T_B = 77.3 K, respectively.

  • (a) If heat is extracted from reservoir A at a rate of $\dot{Q}$ = 1000 W, what is the maximum rate of work that can be generated ($\dot{W}_{\text{max} }$)?

  • (b) If no work is produced, what is the rate of entropy generation ($\dot{S}_{\text{gen}}$) of the cyclic machine?

  • (c) Plot $\dot{S}_{\text{gen}}$ versus $\dot{W}$ (the power produced).

• 2.16. An engineer claimed that it requires much more work to remove 0.1 J of heat from a cryogenic chamber at an absolute temperature of 0.1 K than to remove 270 J of heat from a refrigerator at 270 K. Assuming that the environment is at 300 K, justify this claim by calculating the minimum work required for each refrigeration task.

• 2.17. A solid block [m = 10 kg and c_p = 0.5 kJ/kg K], initially at room temperature ($T_{{{\text{A}},1}}^{{}}$ = 300 K) is cooled with a large tank of liquid–gas mixture of nitrogen at $T_{\text{B}}$ = 77.3 K and atmospheric pressure.

  • (a) After the block reaches the liquid nitrogen temperature, what is the total entropy generation Sgen?

  • (b) Given the specific enthalpy of evaporation of nitrogen, h_fg = 198.8 kJ/kg, what must be its specific entropy of evaporation s_fg in kJ/kg K, in order for the nitrogen tank to be modeled as a reservoir? Does $h_{\text{fg}} = T_{\text{sat}} \times s_{\text{fg}}$ always hold?

• 2.18. Two same-size solid blocks of the same material are isolated from other systems [specific heat c_p = 2 kJ/kg K; mass m = 5 kg]. Initially, block A is at a temperature $T_{{{\text{A}}1}}$ = 300 K and block B at $T_{{{\text{B}}1}}$ = 1000 K.

  • (a) If the two blocks are put together, what will be the equilibrium temperature (T₂) and how much entropy will be generated (S_gen)?

  • (b) If the two blocks are connected with a cyclic machine, what is the maximum work that can be obtained (W_max)? What would be the final temperature of the blocks (T₃) if the maximum work was obtained?

• 2.19. A rock [density ρ = 2800 kg/m³ and specific heat c_p = 900 J/kg K] of 0.8 m³ is heated to 500 K using solar energy. A heat engine (cyclic machine) receives heat from the rock and rejects heat to the ambient at 290 K. The rock therefore cools down.

  • (a) Find the maximum energy (heat) that the rock can give out.

  • (b) Find the maximum work that can be done by the heat engine, W_max.

  • (c) (c) In an actual process, the final temperature of the rock is 330 K and the work output from the engine is only half of W_max. Determine the entropy generation of the actual process.

• 2.20. Consider three identical solid blocks with a mass of 5 kg each, initially at 300, 600, and 900 K, respectively. The specific heat of the material is c_p = 2000 J/kg K. A cyclic machine is available that can interact only with the three blocks.

  • (a) What is the maximum work that can be produced? What are the final temperatures of each block? Is the final state in equilibrium?

  • (b) If no work is produced, i.e., simply putting the three blocks together, what will be the maximum entropy generation? What will be the final temperature?

  • (c) If the three blocks are allowed to interact via cyclic machine but not with any other systems in the environment, what is the highest temperature that can be reached by one of the blocks?

  • (d) If the three blocks are allowed to interact via cyclic machine but not with any other systems in the environment, what is the lowest temperature that can be reached by one of the blocks?

• 2.21. Electrical power is used to raise the temperature of a 500 kg rock from 25 to 500 °C. The specific heat of the rock material is $c_{p} = 0.85$ kJ/kg K.

  • (a) If the rock is heated directly through resistive (Joule) heating, how much electrical energy is needed? Is this process reversible? If not, how much entropy is generated in this process?

  • (b) By using cyclic devices that can interact with both the rock and the environment at 25 °C, what is the minimum electrical energy required?

• 2.22. An insulated cylinder of 2 m³ is divided into two parts of equal volume by an initially locked piston. Side A contains air at 300 K and 200 kPa; side B contains air at 1500 K and 1 MPa. The piston is now unlocked so that it is free to move and it conducts heat. An equilibrium state is reached between the two sides after a while.

  • (a) Find the masses in both A and B.

  • (b) Find the final temperatures, pressures, and volumes for both A and B.

  • (c) Find the entropy generation in this process.

• 2.23. A piston–cylinder contains 0.56 kg of N₂ gas, initially at 600 K. A cyclic machine receives heat from the cylinder and releases heat to the environment at 300 K. Assume that the specific heat of N₂ is c_p = 1.06 kJ/kg K, and the pressure inside the cylinder is maintained at 100 kPa by the environment. What is the maximum work that can be produced by the machine? What is the thermal efficiency (defined as the ratio of the work output to the heat received)? The thermodynamic efficiency can be defined as the ratio of the actual work produced to the maximum work. Plot the thermodynamic efficiency as a function of the entropy generation. What is the maximum entropy generation?

• 2.24. An airstream [$c_{p}$ = 1 kJ/kg K and M = 29.1 kg/kmol] flows through a power plant. The stream enters a turbine at T₁ = 750 K and P₁ = 6 MPa, and exits at P₂ = 1.2 MPa into a recovery unit, which can exchange heat with the environment at 25 °C and 100 kPa. The stream then exits the recovery unit to the environment. The turbine is thermally insulated and has an efficiency η_t = 0.85.

  • (a) Find the power per unit mass flow rate produced by the turbine.

  • (b) Calculate the entropy generation rate in the turbine.

  • (c) Determine the largest power that can be produced by the recovery unit.

• 2.25. Water flows in a perfectly insulated, steady-state, horizontal duct of variable cross-sectional area. Measurements were taken at two ports, and the data were recorded in a notebook as follows. For port 1, speed $\xi_{1} = 3{\text{ m/s}}$, pressure $P_{1} = 50{\text{ kPa}}$, and temperature $T_{1} = 40\;^\circ {\text{C}}$; for port 2, $\xi_{2} = 5{\text{ m/s}}$ and $P_{2} = 45{\text{ kPa}}$. Some information was accidentally left out by the student taking the notes. Can you determine T2 and the direction of the flow based on the available information? Hint: Model the water as an ideal incompressible liquid with c_p = 4.2 kJ/kg) and specific volume v = 10⁻³ m³/kg.

• 2.26. An insulated rigid vessel contains 0.4 kmol of oxygen at 200 kPa separated by a membrane from 0.6 kmol of carbon dioxide at 400 kPa; both sides are initially at 300 K. The membrane is suddenly broken and, after a while, the mixture comes to a uniform state (equilibrium).

  • (a) Find the final temperature and pressure of the mixture.

  • (b) Determine the entropy generation due to irreversibility.

• 2.27. Pure N₂ and air (21% O₂ and 79% N₂ by volume), both at 298 K and 120 kPa, enter a chamber at a flow rate of 0.1 and 0.3 kmol/s, respectively. The new mixture leaves the chamber at the same temperature and pressure as the incoming streams.

  • (a) What are the mole fractions and the mass fractions of N₂ and O₂ at the exit?

  • (b) Find the enthalpy change in the mixing process. Find the entropy generation rate of the mixing process.

  • (c) Consider a process in which the flow directions are reversed. The chamber now contains necessary devices for the separation, and it may transfer heat to the environment at 298 K. What is the minimum amount of work per unit time needed to operate the separation devices?

• 2.28. A Carnot engine receives energy from a reservoir at T_H and rejects heat to the environment at T₀ via a heat exchanger. The engine works reversibly between T_H and T_L, where T_L is the temperature of the higher temperature side of the heat exchanger. The product of the area and the heat transfer coefficient of the heat exchanger is α. Therefore, the heat that must be rejected to the environment through the heat exchanger is $\dot{Q}_{\text{L}} = \alpha (T_{\text{L}} - T_{0} )$. Given $T_{\text{H}} = 800$ K, $T_{0} = 300$ K, and $\alpha = 2300$ W/K. Determine the value of T_L so that the heat engine will produce maximum work, and calculate the power production and the entropy generation in such a case.

• 2.29. To measure the thermal conductivity, a thin film electric heater is sandwiched between two plates whose sides are well insulated. Each plate has an area of 0.1 m² and a thickness of 0.05 m. The outside of the plates are exposed to air at $T_{\infty } = 25\;^\circ {\text{C}}$ with a convection coefficient of h = 40 W/m² K. The electric power of the heat is 400 W and a thermocouple inserted between the two plates measures a temperature of $T_{1} = 175\;^\circ {\text{C}}$ at steady state. Determine the thermal conductivity of the plate material. Find the total entropy generation rate. Comment on the fraction of entropy generation due to conduction and convection.

• 2.30. An electric current, I = 2 A, passes through a resistive wire of diameter D = 3 mm with a resistivity $r_{\text{e}} = 1.5 \times 10^{ - 4} \, \Omega \;{\text{m}}$. The cable is placed in ambient air at $27\;^\circ {\text{C}}$ with a convection coefficient h = 20 W/m² K. Assume a steady state has been reached and neglect radiation. Determine the radial temperature distribution inside the wire. Determine the volumetric entropy generation rate $\dot{s}_{\text{gen}}$ as a function of radius. Determine the total entropy generation rate per unit length of the cable. Hint: For steady-state conduction, $\dot{s}_{\text{gen}} = \frac{1}{T}\nabla \cdot {\mathbf{q}}^{{\prime \prime }} - \frac{1}{{T^{2} }}{\mathbf{q}}^{{\prime \prime }} \cdot \nabla T$. [Hint: Consider $\kappa = 10{\text{ W/m}}\,{\text{K}}$ and $\kappa = 1{\text{ W/m}}\;{\text{K}} . ]$

• 2.31. Find the thermal conductivity of intrinsic (undoped) silicon, heavily doped silicon, quartz, glass, diamond, graphite, and carbon from 100 to 1000 K from Touloukian and Ho [13]. Discuss the variations between different materials, crystalline structures, and doping concentrations.

• 2.32. Find the thermal conductivity of copper from 1 to 1000 K from Touloukian and Ho [13]. Discuss the general trend in terms of temperature dependence, and comment on the effect of impurities.

• 2.33. For laminar flow over a flat plate, the velocity and thermal boundary layer thicknesses can be calculated by $\delta (x) = 5x/\sqrt {{{Re}}_{x} }$ and $\delta_{\text{t}} (x) = 5x{{Re}}_{x}^{ - 1/2} {Pr }^{ - 1/3}$, respectively. Use room temperature data to calculate and plot the boundary layer thicknesses for air, water, engine oil, and mercury for different values of $U_{\infty }$. Discuss the main features. Hint: Property data can be found from Incropera and DeWitt [11].

• 2.34. Air at 14 °C and atmospheric pressure is in parallel flow over a flat plate of $2 \times 2{\text{ m}}^{ 2}$. The air velocity is 3 m/s, and the surface is maintained at 140 °C. Determine the average convection coefficient and the rate of heat transfer from the plate to air. (For air at 350 K, which is the average temperature between the surface and fluid, $\kappa = 0.03{\text{ W/m}}\;{\text{K}}$, $\nu = 20.9 \times 10^{ - 6} {\text{ m}}^{ 2} / {\text{s}}$, and ${Pr}\; = \;0.7$.)

• 2.35. Plot the blackbody intensity (Planck’s law) as a function of wavelength for several temperatures. Discuss the main features of this function. Show that in the long-wavelength limit, the blackbody function can be approximated by $e_{\text{b},\lambda } (\lambda ,T) \approx 2\pi ck_{\text{B}} T/\lambda^{4}$, which is the Rayleigh–Jeans formula.

• 2.36. Calculate the net radiative heat flux from the human body at a surface temperature of $T_{\text{s}} = 308{\text{ K}}$, with an emissivity $\varepsilon = 0.9$, to the room walls at 298 K. Assume air is at 298 K and has a natural convection coefficient of 5 W/m² K. Neglect evaporation, calculate the natural convection heat flux from the person to air. Comment on the significance of thermal radiation.

• 2.37. Combustion occurs in a spherical enclosure of diameter D = 50 cm with a constant wall temperature of 600 K. The temperature of the combustion gases may be approximated as uniform at 2300 K. The absorption coefficient of the gas mixture is $a_{\lambda } = 0.01{\text{ cm}}^{ - 1}$, which is independent of wavelength. Assuming that the wall is black and neglecting the scattering effect, determine the net heat transfer rate between the gas and the inner wall of the sphere.