LECTURES IN THERMODYNAMICS Claus Borgnakke CHAPTER 2 For the 8th Edition of: Fundamentals of Thermodynamics Claus Borgnakke, Richard Sonntag John Wiley & Sons, 2013

Chapter 2 Phase Boundaries and The P-v-T Surface The Two Phase States The Liquid and Solid States The Superheated Vapor States The Ideal gas States Compressibility Factor Equations Of State (EOS) Computerized Tables

Phase Boundaries and The P-v-T Surface Measure P-v-T and observe phase for water. Water at 1 atm (101 kPa), room temperature is liquid. Heat the water, state moves from (a) towards (b) . At (b) it is boiling with co-existing liquid and vapor. Continued heating: more vapor, less liquid until no more liquid. Further heating gives warm vapor at (c). If water is cooled it will change state from (a) towards (e) and it will start to freeze at (d). Further cooling gives solid water (ice) with no liquid at (e). If process is repeated at different pressures we can indicate the phases in a P-T diagram, where the points for the phase changes are connected with red curves. On these lines points are pairs of saturated (P, T) a b c d e

Phase Boundaries and The P-v-T Surface Complete phase diagram (P, T) separates the solid (S), the liquid (L) and the vapor (V) phases. Sublimation line (S above, V below) Fusion line (S left, L right) Vaporization curve (L above, V below) On each curve: P = Psat(T) or T = Tsat(P) One point on each curve is a pair (Psat, Tsat) State a: Superheated vapor (T > Tsat for same P) Expanded vapor (P < Psat for same T) State b: Compressed liquid (P > Psat for same T) Subcooled liquid (T < Tsat for same P) Vaporization curve

Phase Boundaries and The P-v-T Surface The phase diagram for water Notice the P axis must be log(P) to fit the large range. The phase diagram for CO2

Phase Boundaries Concept Questions

The P-v-T surfaces Show the (T, v) combinations following constant P processes for water Small v’s are liquid Large v’s are vapor (gas) Liquid and vapor regions are separated by a 2-phase region of mixtures with a combination of liquid and vapor. The borders are the Saturated liquid line and Saturated vapor line They meet at critical point vapor liquid

The P-v-T surface for water The total P-v-T surface shows all the different regions Small v’s are liquid or solid Large v’s are vapor (gas) There are 3 different 2-phase regions: Solid-Vapor Solid-Liquid Liquid-Vapor Look at this diagram from the right along the volume axis. You see the P-T phase diagram. Look from the top down, you see the T-v diagram. Look from the left along the T axis you see a P-v diagram partly obscured by the solid surface. We will use these 2-D figures/projections (for simplicity)

The P-v-T surface for most substances other than water Water has a higher v for solid than liquid so the solid is lighter than the liquid Most other substances behaves opposite Notice also that the fusion line separating the solid and liquid region slopes positive Recall the phase diagram for CO2 All the two-phase surfaces are plane in the v direction. When you look from the left along the v axis the planes show up as curves: Sublimation, Fusion and Vaporization lines

The B Section Tables for Water Saturated two-phase liquid + vapor region Table B.1.1 Gives listing of Tentry Psat, vf, vfg, vg, …. Table B.1.2 same but with Pentry Tsat, … vf vfg vg Psat as function of T

The Tables for Water Table B.1.3 superheated vapor along a constant P curve: Tentry, v, … Table B.1.4 compressed liquid Subheading P (Tsat) TABLE B.1.3 Superheated Vapor v u h s

The Tables for Water The saturated solid-vapor table and the sublimation line Table B.1.5 Gives listing of Tentry Psat, vi, vig, vg, ….

The Tables for Water

The Two Phase States

The Tables for Water

The Tables for Water Table B.5.1 T Psat vf vfg vg uf ufg ug

The Tables for Water Table F.10.1 T Psat vf vfg vg uf ufg ug

The Liquid and Solid States Main characteristic: Incompressible, weak function of T, meaning v ≈ constant = 1/ρ = v(T) Solid: v = v(T) ≈ vi Liquid: v = v(T) ≈ vf If P > Psat compressed solid If P > Psat compressed liquid Figures illustrate properties for water

The Superheated Vapor States You have a superheated vapor when v is larger than the saturated vapor value v > vg The region above the critical point is called dense fluid.

The Superheated Vapor States

The Superheated Vapor States

The Superheated Vapor States Example 2.4 The red circles indicates the two states used for the interpolation

The Superheated Vapor States Example 2.4E The red circles indicates the two states used for the interpolation

The Superheated Vapor States Example 2.5

The Superheated Vapor States Example 2.5 continued From B.1.3: Locate state 200 C, v = 0.4 m3/kg 500 kPa, 200 C: v = 0.42492 m3/kg; 600 kPa, 200 C: v = 0.35202 m3/kg

The P-v-T Surface Concept Questions

The Superheated Vapor States and The Ideal Gas States Constant v, T and P curves are shown below. When extended far out in the superheated vapor region they become simple (math) curves. P = BT for constant v; Pv = C for constant T; T = Av for constant P B = Bo / v C = Co T A = Ao P Unified equation: Pv = RT Ideal gas law, R ideal gas constant R = R /M R : universal gas constant M: molecular mass L V L + V S

Ideal Gas Law P v = R T P V = m R T = n R T A different scaling can be used P v = R T P V = m R T = n R T Here n is the number of moles and M is the molecular mass. n = m / M From ideal gas law: 1 mol occupies same V at a given (P, T) regardless of which substance it is. Compare a mass at state 1 with same mass at a state 2: m R = P2V2 T2 = P1V1 T1 Differentiate the ideal gas law with time for constant state (P, T ) to get:

Ideal Gas Law Example 2.6

Ideal Gas Law Example 2.7 Since substance not given R is calculated from universal gas constant, recall 1 kN m = 1 kJ.

Ideal Gas Law Example 2.7E Since substance not given R is calculated from universal gas constant.

Ideal Gas Law Example 2.8 Differentiate the ideal gas law with time at constant ( P, T ) to get:

Ideal Gas Law When do you have an ideal gas? For very large v, so molecular distances are large (very small force between molecules) To get a sense look at a T-v diagram for water. If T is higher you can allow smaller v’s and still have an ideal gas.

The Compressibility Factor The deviation from the ideal gas law can be expressed by a factor Z so P v = Z RT For an ideal gas Z = 1 and for states approaching liquid Z becomes very small. For very high pressures Z can actually be higher than 1. A chart for nitrogen gives a general shape of a compressibility chart.

The Compressibility Factor The compressibility chart shows Z plotted as an average chart in reduced properties. The scaling is done with the critical point properties: Pr = P/Pc Tr = T/Tc The chart is plotted using an equation of state called Lee Kesler.

The Compressibility Factor Example 2.11

The Compressibility Factor Example 2.11 continued

The Ideal Gas and Compressibility Concept Questions

Equations of State P = RT v – b – a v2 + cbv + db2 The P-v-T surface is approximated in an equation of state. Several of these are extensions of the ideal gas law as P = RT v – b – a v2 + cbv + db2 Here a and b are scaled to the critical properties a = ao Pc x2 ; b = bo x ; x = RTc/Pc c and d are model constants shown in Tabel. D.1 including a correction with the accentric factor, ω

Computerized Tables CATT3 CATT3 program with water selected. Shows the T-s or the P-v (log-log) diagrams. L + V two-phase region purple, superheated vapor red and liquid, dense fluid blue, green/aqua State selected (100 oC, 0.1 MPa) shown by crosshair lines.