Chemical Process calculation III

Transcription

1 Chapter 7 Ideal and Real Gases Gas, Liquid, and Solid Chemical Process calculation III Gas: a substance in a form like air, relatively low in density and viscosity Liquid: a substance that flows freely like water Solid: a substance that is hard, has a definite shape, and relatively high in density Gas and liquid conform to the shape of a container, but solid does not. Gas and Vapor The vapor in its natural state is a solid or a liquid at room temperature. Ex) water vapor, alcohol vapor, nitrogen gas Ideal gas: a gas that has negligible molecular volume and intermolecular forces, obeying the ideal gas law. PV = nrt, which is called the equation of state for ideal gases. where P is the absolute pressure of the gas, V is the volume occupied by the gas, n is the number of moles of the gas, and R is the ideal gas constant, and T is the absolute pressure of the gas. Standard conditions: T = K, P = kpa or 1 atm, V = L/mol The gas constant can be calculated using the ideal gas law with the given standard conditions. R The values of R in other units: 1 atm L L atm K mol mol K

2 Example 7.1 Example 7.2 Density and specific gravity: Density = where M is the molecular weight. Specific gravity (sp. gr.) = Idea gas mixture: P p where P is the total pressure and pi is the partial pressure of component i in a mixture. p V n RT PV = nrt p n n P y P Example 7.3 Example 7.4 Example 7.5 Real gases Table 7.4 Equations of state for real gases. van der Waals, Soave-Redlich-Kwing (SRK), Peng-Robinson (PR), Redlich-Kwong (RK), Benedict-Webb-Rubin (BMW), Kammerlingh-Onnes, and Holborn equation. Keep in mind that you must know the region of validity of any equation of state and not extrapolate outside the region, particularly not into the liquid region, by ignoring the possibility of condensation for gases. Example The critical state and compressibility As the temperature is increased or the pressure is increased, the density of a substance in liquid state approaches to that in vapor state. Eventually, a critical state is reached where the densities of the liquid and vapor become identical.

3 Figure 7.5 Pressure versus Specific Volume. The vertical axis represents pressure and horizontal axis represents specific volume. A supercritical fluid is a compound in a state above its critical point. Reduced variables: Reduced temperature, T Reduced pressure, P Reduce specific volume, V Compressibility factor, z = PV znrt z equals one for ideal gases or real gases that behave like ideal gases. Fug. 7.6 (a) Compressibility at 100 o C for several gases as a function of pressure; (b)

4 compressibility factor for several gases as a function of reduced pressure and reduced temperature. z is a function of T r and P r, and can be represented by the following equation: z z z ω wheer ω is the acentric factor, which indicates the degree of acentricity or nonsphericity of a molecule. z 0 and z 1 are given in Appendix, and ω is shown in Table 7.3 Group contribution method: This method is based on combining contribution of each functional group of a compound. The key assumption is that a group such as CH 3 or OH behaves identically irrespective of the molecule in which h it appears. UNIFAC and UNIQUAC are popular group contribution methods, which form a part of many computer databases. The group contribution methods are used not only for the compressibility factor but also for other physical properties. Chapter 8 Multiphase Equilibrium A substance can exist in many phases: vapor, liquid and solid. Phase diagram: A phase diagram is a chart to show the phase of a substance at a given condition (pressure, temperature, volume, etc.) 2 dimensional phase diagrams (pressure-temperature, pressure-volume, etc.) are used commonly, although 3 dimensional phase diagrams are available.

5 Figure 8.4 Pressure-volume phase diagram. Vapor pressure of a substance at a given temperature is defined as the pressure at which the vapor is in equilibrium with the liquid or with the solid. The vapor pressure increases with increasing temperature. A liquid boils when the vapor pressure equals the system pressure under which the liquid is located. Triple point: the point at which liquid, solid, and vapor coexist in equilibrium. Vaporization: the change of phase from liquid to vapor Condensation: the change of phase from vapor to liquid Sublimation: the change of phase from solid to vapor Bubble point: the temperature at which a liquid just starts to vaporize Dew point: the temperature at which a vapor just starts to condense Freezing (solidifying): the change of phase from liquid to solid Melting (fusion): the change of phase from solid to liquid Saturated liquid or vapor: liquid or vapor in equilibrium with the other Subcooled liquid: liquid at a temperature lower the saturated liquid Superheated vapor: vapor at a temperature higher than the saturated vapor Supercritical: above the critical point or higher than critical temperature and pressure In a supercritical region, a substance is neither a gas nor a liquid, but a supercritical fluid, which exhibits properties that lie between the gas and the liquid. Phase Rule: F = 2 P + C F: number of degrees of freedom (number of intensive properties that can be varied independently) P: number of phases C: number of components Intensive property: properties that do not depend on the quantity of material present. (pressure, temperature, concentration, density, specific heat, specific gravity, etc.) Extensive property: properties that do depend on the quantity of material present. (volume, mass, mole, energy, etc.) How many degrees of freedom?

6 For a pure gas: F = =2 Two properties can be varied independently, temperature and pressure, temperature and specific volume, or pressure and specific volume. The other property is not independent, but determined from the two independent ones, using the gas law. For a pure gas in equilibrium with its vapor: F = = 1 When water boils, liquid and vapor are in equilibrium. Either the temperature of the pressure can be varied independently, but both of them cannot be varied independently. If the boiling pressure is specified, the boiling temperature is not variable, but fixed. Example 8.1 Relationship between vapor pressure and temperature: Antoine equation: ln where A, B, C = constants for each substance T = temperature, K B C T Vapor pressures at various temperatures are fitted by the equation to determine the three constants, A, B and C. Steam table (attached foldout) 1. Enthalpy vs. temperature 2. Properties of superheated steam (specific volume and enthalpy) as a function of temperature and pressure 3. Properties of subcooled water (density, enthalpy and specific internal energy) as a function of temperature and pressure. Example) A 10 m 3 vessel contains 2000 kg of water plus saturated steam at 10 atm. What is the temperature? Determine the volumes and the masses of water and steam. Interpolating in the steam tables: Estimation of vapor pressure at a temperature between two temperatures of vapor pressure: T (K) p* (kpa) Estimate the vapor pressure at 312 K by linear interpolation. 8.4 Two-component gas/single-component liquid system Ex) Water vapor and a non-condensible gas (air) over liquid water.

7 When a liquid is put in a closed container, the liquid vaporizes until the partial pressure reaches the vapor pressure of the liquid. Either a further decrease in the temperature or a further increase in the partial pressure brings a condensation of the vapor. At the dew point Ex) Suppose that you have an air saturated with water at 51 o C in a container at 750 mmhg. - What is the vapor pressure of water? - What is the partial pressure of water vapor? - What is the mole fraction of water vapor? - What is the mole fraction of air? - How could you condense the water vapor? Example 8.5) Initial condition: Saturated air at 75 o F and 1 atm Determine the partial pressure and the mole fraction of the water vapor. Change 1: The saturated air is compressed to 4 atm. Determine the amount of water of water vapor that was condensed. Change 2: Remove the liquid water and expand the air isothermally back to 1 atm. Determine the partial pressure of the water vapor and the dew point temperature. Example 8.6 Example 8.7 Two-Component Gas/Two-Component Liquid System Ex) water-alcohol

8 How to predict the partial pressure of a component i in a mixture? For ideal solutions, p x P (Raoult s law) where p = partial pressure of component i in the vapor phase x = mole fraction of component i in the liquid phase P = vapor pressure of pure component i at the given temperature The Raoult s applies for solutions quite similar in chemical nature or for a component whose mole fraction in the liquid phase approaches to 1.0. For a component whose mole fraction in the liquid phase approaches to 0, the following Henry s law applies. p H x where Hi is the Henry s law constant for component i. Take, for example, CO 2 dissolved in water at 40 o C. The partial pressure of the carbon dioxide is 1 atm. The Henry s law constant for CO 2 is 69,600 atm/mole fraction. Calculate the mole fraction of CO 2 in the liquid phase. Calculate the partial pressure of CO 2 vapor in equilibrium with 4.2 x 10-6 mole fraction of CO 2 in the liquid phase. Temperature and composition or temperature and pressure or pressure and composition can be varied independently. Phase diagrams for a mixture of benzene and toluene:

9 What is the mole fraction of toluene at the initial state? Determine the dew point pressure or the pressure at which the condensation begins. What is the mole fraction of toluene in the first condensate? The vapor is enriched in benzene as the condensation proceeds. Determine the mole fraction of toluene in the condensate at the benzene mole fraction of 0.75 in the vapor. Figure 8.19 shows a phase diagram of a mixture of isopropanol and water at 1 atm where an azeotrope is displayed. At the azeotropic point, the mole fraction of vapor is equal to the mole fraction of liquid and no further enrichment in specific component is possible. K-value (vapor-liquid equilibrium ratio) K y x

10 The higher the K value, the more volatile a component. Dew point and bubble point calculations: At the bubble point, the partial pressures (p i ) of components add up to the system pressure (P). P p γ x P (for ideal solutions, γ 1.0) Determine the bubble point temperature by trial and error so the above requirement could be met. At the dew point, the mole fractions in the liquid phase add up to 1.0 p 1.0 γ P Determine the dew point temperature by trial and error so the above requirement could be met. Example 8.9