Thermodynamics of Multiphase Systems

Abstract

This chapter presents the thermodynamics of multiphase systems, which begins with the fundamentals of equilibrium and stability. This is followed by a discussion of multicomponent multiphase systems and the metastable equilibrium that exists in the multiphase system. This chapter concludes with a discussion of thermodynamics at the interface, surface tension, disjoining pressure and superheat.

Problems

• 2.1. Show that the equilibrium criterion for a system with constant entropy and pressure is $\Delta H_{S,p} \le 0$.

• 2.2. A simple system is confined by an adiabatic, rigid, and impermeable boundary. In its initial state, the system is divided by a diathermal partition (a partition that allows heat to penetrate) into two equal halves. The internal energies of the left and right halves satisfy

  $$E_{\text{L}} = \frac{3}{2}n_{\text{L}} R_{\text{u}} T_{\text{L}} \quad E_{\text{R}} = \frac{3}{2}n_{\text{R}} R_{\text{u}} T_{R}$$

  where the mole numbers of the left and right sides are n_L = 2 kmol and n_R = 3 kmol, respectively. The initial temperatures of the two halves are, respectively, T_L = 250 K and T_R = 350 K. After thermal equilibrium has been established, find (a) the values of E_L and E_R, and (b) the equilibrium temperature.

• 2.3. Two cylinders are fitted with two pistons coupled together so that the linear displacement of the two pistons must be the same (see Fig. P2.3). The ratio of the cross-sectional area of the two cylinders is A_L/A_R = 0.5. A bar with high thermal conductivity connects the two cylinders so that the final temperatures of the two cylinders are the same. Determine the ratio of the pressures in the two cylinders, $p_{\text{L}} /p_{\text{R}}$, at equilibrium.

  

  Fig.  P2.3

• 2.4. A closed system is initially divided into two halves. The pressures of the two halves are identically equal to p. The initial entropies of the left and right sides are both equal to S, and the total entropy of the system is 2S. After an infinitesimal change, the entropy of the left and right halves becomes $S_{\text{L}} = S + \Delta S$ and $S_{\text{R}} = S - \Delta S$. The pressures of the two halves remain unchanged throughout the process. Using the enthalpy minimum principle, show that the system is stable if the specific heat at constant pressure is positive, $c_{\text{p}} > 0$.

• 2.5. Show that the specific heat at constant pressure is always greater than that at constant volume for any substance, i.e., $c_{\text{p}} > c_{\text{v}}$.

• 2.6. The following two equations are valid for an ideal gas:

  $$pV = nR_{\text{u}} T\quad E = cnR_{\text{u}} T$$

  where $R_{\text{u}}$ is the universal gas constant and c is a constant which depends on the molecular structure of the ideal gas. What is the entropy of the ideal gas?

• 2.7. A 1-m³ vessel is filled with propane at room temperature of 20 °C and pressure of 100 kPa. Find the mass of the propane by using (a) the ideal gas law and (b) the van der Waals equation.

• 2.8. Reduced pressure, temperature, and specific volume can be defined as

  $$p_{\text{r}} = \frac{p}{{p_{\text{c}} }}\quad T_{\text{r}} = \frac{T}{{T_{\text{c}} }}\quad v_{\text{r}} = \frac{v}{{v_{\text{c}} }}$$

  where p_c, T_c, and v_c are the pressure, temperature, and specific volume at the critical point. Show that the van der Waals equation in terms of reduced pressure, temperature, and specific volume is

  $$p_{\text{r}} = \frac{{8T_{\text{r}} }}{{3v_{\text{r}} - 1}} - \frac{3}{{v_{\text{r}}^{2} }}$$

• 2.9. For a fluid that does not satisfy the ideal gas law, a compressibility factor Z is introduced: $Z = pv/RT$; this can also be written in terms of reduced pressure, temperature, and specific volume, i.e., $Z = Z_{\text{c}} p_{\text{r}} v_{\text{r}} /T_{\text{r}}$, where Z_c is the compressibility factor at the critical point. Determine the compressibility of the fluid that satisfies the van der Waals equation of state.

• 2.10. For a gas that satisfies the van der Waals equation, show that its internal energy can be expressed as

  $$e = e_{0} + c_{\text{v}} (T - T_{0} ) + a\left( {\frac{1}{{v_{0} }} - \frac{1}{v}} \right)$$

  if the specific heat at constant volume, c_v, is constant. The subscript 0 in the equation denotes a reference state.

• 2.11. Redlich–Kwong equation is generally considered to be the best among the two-constant equations of state:

  $$p = \frac{{R_{\text{g}} T}}{v - b} - \frac{a}{{v(v + b)T^{1/2} }}$$

  Obtain the two constants a and b in the Redlich–Kwong equation by applying the critical point conditions: $(\partial p/\partial v)_{T} = 0$ and $(\partial^{2} p/\partial v^{2} )_{T} = 0.$

• 2.12. Using the Clapeyron equation (2.156), estimate the value of the latent heat of vaporization of water at 100 °C, and compare it with the value from Table B.48.

• 2.13. Determine the boiling point of water in a city where the elevation is 1500 m. The atmospheric pressure at this elevation is 84.56 kPa.

• 2.14. The temperature of ice in an ice rink is −5 °C. An ice skater is standing on one foot, and the contact area between the skate and ice is 280 × 2 mm². The weight of the ice skater is 60 kg. What is the melting point of the ice underneath the skate?

• 2.15. The saturation temperature of water at 1 atm is 100 °C. Use the Clausius–Clapeyron equation to find the saturation pressure if the temperature is increased to 110 °C and compare your result with that obtained by using a steam table

• 2.16. A system with constant temperature and pressure contains two phases of the same substance. Show that the two phases in the system are in equilibrium if Eqs. (2.126)–(2.128) are satisfied.

• 2.17. The chemical potential of a single-component system is its Gibbs free energy, $\bar{g}$ [see Eq. (2.146)]. According to Eq. (2.128), the equilibrium chemical potential of the liquid and vapor at the liquid–vapor interface must be equal: $g_{\text{v}} = g_{\ell }$. Start from this relation, and show that the latent heat of vaporization is $h_{{\ell {\text{v}}}} = T(s_{\text{v}} - s_{\ell } )$ where $s_{\text{v}}$ and $s_{\ell }$ are the entropies of saturated vapor and liquid, respectively.

• 2.18. A mixture of saturated liquid and vapor is in a piston–cylinder system as shown in Fig. P2.18. The piston is frictionless, so the piston–cylinder system is a system with constant pressure and temperature. If the masses of liquid and vapor are, respectively, $m_{\ell } {\text{ and }}m_{\text{v}}$, the Gibbs free energy of the mixture is $G = m_{\ell } g_{\ell } + m_{\text{v}} g_{\text{v}}$. Show that the condition for the liquid and vapor phases at phase equilibrium is $g_{\ell } = g_{\text{v}}$.

  

  Fig. P2.18

  

  Fig.  P2.20

• 2.19. A rigid tank filled with saturated nitrogen vapor at $T = 100\,{\text{K}}$ is cooled to condense the vapor. It is assumed that the van der Waals equation is valid and the two constants are $a = 167.15\,{\text{Pa m}}^{6} / {\text{kg}}^{2}$ and $b = 1.35 \times 10^{ - 3} \,{\text{m}}^{3}$. The system is in a metastable state before condensation starts. Determine the temperature and the corresponding pressure at which condensation will occur.

• 2.20. A mixture of liquid and vapor water fills a chamber with the wall temperature maintained at a constant level, T_w (see Fig. P2.20). A valve near the bottom of the chamber is opened, and 1 kg of water is drained from the chamber. Find an expression, in terms of the saturation properties of liquid and vapor, for the increase in the volume of the vapor phase.

• 2.21. Surface tension has been represented in terms of internal energy [Eq. (2.203)], Helmholtz free energy [Eq. (2.208)], and Gibbs free energy [Eq. (2.220)]. Find the representation of the surface tension using enthalpy.

• 2.22. The degree of supersaturation was expressed in terms of the ratio of the vapor pressure and saturation pressure in Example 2.7. Express the degree of supersaturation in Example 2.7 in terms of temperature using the Clapeyron–Clausius equation.