CHAPTER 10 PROBLEMS

J. CHAPTER 10 PROBLEMS

Problem J1 If you strike a match in a class room, it may not result in explosion; but if you strike (never try this) a match near a gas station, there may be an explosion. Relate this to stability.

Problem J2 If there can be phase change (i.e., the formation of two regions with two different densities) at given T and P, why cannot there be different thermal layers at specified values of ρ and P?

Problem J3 Derive an expression for the spinodal condition for a fluid following the Peng Robin- son equation of state. Obtain the spinodal curve for both liquid and vapor n–hexane and plot P(V), P(T), T(v) at specified T, v and P respectively.

Problem J4 Plot P(v) for H 2 O at 373 K using the RK state equation for volumes in the range 0.0867 v c ’ to 200 v c ’, v c ’= R T c /P c . If g (373K, 200 v c ’) = 0 and dg = vdP, plot g(v) at 373 K. (Hint: use integration by parts.) Determine the states at which g liquid =g vapor , and at which dP/dv = 0.

Problem J5 Consider the relation G–A = PV. Since d 2 G < 0 at constant values of T and P for a single component fluid, show that d 2 A < –P d 2 V at these values of T and P.

Problem J6 Refrigerant R–134A enters a short 1 mm diameter and 12 mm long capillary tube at 6 bar and with 30ºC subcooling (i.e., it is 30ºC below its saturation temperature at 6 bar). The fluid is discharged into a 1 bar atmosphere. The pressure first decreases rapidly due to vena contracta (expansion in a converging) and then rises due to the in- creased diameter of the flow (as in a diffuser). At a certain pressure R–12 starts flashing (or vaporizing). Since its density is low, the flow velocity rapidly increases with a subsequent decrease in pressure and may reach the sound speed (or choked flow condition). Data is needed on spinodal conditions. Determine the spinodal pres- sures at 30ºC, 20ºC, and 10ºC subcooled conditions. Use Dietrici equation of state. How does this compare with values determined from the charts for the RK equation.

Problem J7 Obtain an expression for vapor spinodal curve for both P and T with respect to v as- suming that b«v. Use the Berthelot equation.

Problem J8 Prove that if S = S (U, V, N 1 ,N 2 , ..., N n ) at equilibrium, and if dS < 0 due to a pertur- bation, then for the perturbed state dH T,S > 0, dA T,V > 0, and dG T,P > 0.

Problem J9 If u = u(s,v), show that the stability criteria for u min are c v >0 and ( ∂P/∂v) s <0.

Problem J10 Starting with S–S e = ((U–U e ) /T) + (P/T) (V–V e ) + C and C < 0, show that (a) at specified values of S, V and m, (b) U > U e , at specified values of T, V, and m, A>A e ,

and c) at specified values of T, P and m, G > G e .

Problem J11 Obtain the spinodal condition for the state equation P = RT/(v–b) – a/(T n v 2 ).

Problem J12 Obtain the spinodal curve for a RK gas in terms of P R vs. T R and P R vs. v ′ R .

Problem J13 Determine the maximum temperatures to which liquid water can be superheated and vapor can be subcooled at 133 bar.

Problem J14

2 ∂ 2 S= 2 S/ ∂U 2 dU 2 + ∂ 2 2 2 2 Show that d 2 S/ ∂V dV +2 ∂ S/ ∂U∂V dU dV = –(C v /T ) dT + ( ∂P/∂V)

2 T (1/T) dV . Problem J15

For a VW gas s = c vo ln ((u+a/v)/c v0 ) + R ln (v–b). Using the criterion ∂ 2 s/ ∂v 2 < 0, obtain the following expression for values of v which satisfy above criteria at speci-

fied u, namely, (k–1) (1 + x) 2 /(1– b * x) 2 > x (2+x), where x = a/(vu), b * = (bu)/a, and k =c po /c vo .

Problem J16 Using the Berthelot equation of state P = RT/(v–b) – a/(Tv 2 ) for water plot T(v) for P

= 1, 10, 20, 40, 60, 80, 100, 200 bar. At P = 60 bar determine the maximum tempera- ture to which water can be superheated without forming vapor and the minimum tem- perature to which water can be cooled without causing condensation. Assume that ln

P ,sat R ≈7(1– 1/T R ). Plot the saturated liquid and vapor curves and determine the degree of superheat and subcooling at 60 bar. Problem J17 Consider n-hexane (C 6 H 14 ) at P=4 bar, v= 0.3m3/kmol following Van der Waals

equation of state. Verify whether n-hexane is mechanically stable at this condition. Problem J18

For the state equation P = RT/(v–b) – a/(T n v m ) for what values of m is it possible to obtain the spinodal conditions for T<T c ? Can we relate this to the universe, treating galaxies as point masses?

Problem J19 In the context of the Peng Robinson state equation solve for T(P,v). Plot T with re- spect to P at 1 bar and 60 bar. Determine the degree of superheat and subcooling.

Problem J20 Use the VW equation of state P = RT/(v–b) – a/v 2 for H

2 O. Plot P(v) at 593 K. Recall the expression s = c vo ln ((u +a/v)/c vo ) + R ln (v–b), where T = (u +a/v)/c

vo 2 . Plot s(v, 593 K), assuming that c vo = 54 kJ kmol K . Determine d s along the 593 K iso- therm. Discuss your results in the context of stability criteria.

Problem J21 Saturated liquid water (the mother phase) is kept in a piston cylinder assembly at a pressure of 100 kPa. A minute amount of heat is added to form a single vapor bubble (the embryo phase). a) If the embryo phase is assumed to be at the same temperature and pressure as the mother phase, determine the absolute stream availabilities ψ=h–

T 0 s and Gibbs functions of the mother and embryo phases. b) If the embryo vapor phase is at the spinodal pressure corresponding to 100ºC while the liquid mother phase is still at 1 bar, what are the absolute stream availability and Gibbs function of the vapor embryo? Compare the answers from parts (a) and (b). For the spinodal pres- sure assume that the RK equation applies. (In order to use the values from saturation tables assume that the vapor phase behaves as an ideal gas to calculate the enthalpies and entropies between the saturated and spinodal states.) c) If the embryo at the spi- nodal pressure condenses back to the mother liquid phase at 100ºC and 1 bar what is the change in the Gibbs function?

Problem J22 Obtain the stability criteria for an ideal gas using the criteria related to h PP ,h ss , and

h sP . Apply the relations dh = c po dT and s = c po dT/T – R dP/P. Problem J23

Show that u

ss u vv -u sv = -a vv /a TT .