CHAPTER 7 PROBLEMS

G. CHAPTER 7 PROBLEMS

Problem G1

For an ideal gas c vo =c vo (T) and, hence, u o =u o (T). Is this true for a VW gas?

Problem G2 How will you analyze transient flow processes discussed in example 14 of Chapter 2 for real gases?

Problem G3 Recall that du T = (a/T v 2 ) dv for a Berthelot gas. The integration constant F(T) can

be evaluated at the condition a →0. Is the expression for F(T) identical to that for an ideal gas? Problem G4 If, for a gas, du = c v dT + f(T,v)dv and c v =c v (T,v), which is unknown, can we deter-

mine the value of u by integrating the expression at constant values of v? Problem G5

Is it possible to predict the properties s fg , and h fg using “real gas” state equations? Problem G6

An insulated metal bar of cross sectional area A is stretched through a length dx by applying a pressure P. Does the bar always cool or heat during this process?

Problem G7 The residual internal energy of a Berthelot fluid u(T,v) – u o (T) = –2a/(Tv). Determine an expression for the residual specific heat at constant volume c v (T,v) – c vo (T).

Problem G8

A rubber product contracts upon heating in the atmosphere. Does the entropy increase or decrease if the product is isothermally compressed? (Hint: Use the Maxwell’s rela- tions.)

Problem G9

a) Using the generalized thermodynamic relation for du, derive an expression for u R /RT c for a Clausius II fluid. b) What is the relation for c vo (T)– c v (T,v) for the fluid? c)Determine the values of u R /RT c and h R /RT c for CO 2 at 425 K and 350 bar.

Problem G10

Determine an expression for ∂c v / ∂v for a Clausius II fluid in terms of v and T. Problem G11

Consider the isothermal reversible compression of Ar gas at 180 K from 29 bar to 98 bar in a steady state steady flow device. Using the fugacity charts determine the work in kJ per kmol of Ar.

Problem G12 Assume that air is a single component fluid. Air is throttled in order to cool it to a temperature at which oxygen condenses out as a liquid.

a) In order to determine the inlet conditions for the throttling process you are asked to determine the inversion point. Looking at the charts presented in text for RK equation, determine the inversion pressure at 145.38 K.

b) Using the c v relations, determine c v of air at the inversion point. c)

Determine c p at this inversion condition. Assume that c p o = 29 kJ kmol –1 . Is

the value of c p – c v = R ?

d) What is the value of the Joule Thomson Coefficient at 1.2 times the inver- sion pressure at 145.38 K. Assume that the value of c p at this pressure

equals that at the inversion point. Do you believe air will be cooled at this point?

Problem G13 Near 1 atm, the Berthelot equation has been shown to have the approximate form Pv = RT (1 + (9PT

c /(128P c T)) 1–6(T c /T ))). Obtain an expression for s(T,P). Problem G14

In a photon gas the radiation energy is carried by photons, which are particles without mass but carry energy. The gas behaves according to the state equation P = 4 σT 4 /(3

c 0 ), where σ denotes the Stefan Boltzmann constant and c o the light speed in vacuum. Obtain an expression for the internal energy by applying the relation du = c v dT +

(T( ∂P/∂T) v – P) dv. Problem G15

Oxygen enters an adiabatic turbine operating at steady state at 152 bar and 309 K and exits at 76 bar and 278 K. Determine the work done using the Kessler charts. Ignore Pitzer effects. What will be the work for the same conditions if a Piston-cylinder sys- tem is used?

Problem G16 The Joule Thomson effect can be depicted through a porous plug experiment that il- lustrates that the enthalpy remains constant during a throttling process. In the experi- ment a cylinder is divided into two adiabatic variable volume chambers A and B by a rigid porous material placed between them. The chamber pressures are maintained constant by adjusting the volume. Freon vapor with an initial volume V A,1 , pressure P A,1 and energy U A,1 is present in chamber A. The vapors penetrate through the porous wall to reach chamber B. The final volume of chamber A is zero. Determine the work done by the gas in chamber B, and the work done on chamber A. Apply the First Law for the combined system A and B and show that the enthalpy in the combined system is constant.

Problem G17 Obtain a relation for the Joule Thomson coefficients for a VW gas and an RK gas in terms of a, b, c p , R, and T. Determine the inversion temperature.

Problem G18 Obtain an expression for f/P for a VW gas and write down the expression at the criti- cal point. Assume that the gas behaves like an ideal gas at a low pressure P o and large

volume v o . (Hint: ∫vdP = Pv– ∫Pdv, and P 0 v 0 = RT.)

Problem G19

The Cox–Antoine equation is ln P = A – B/(T+C). Determine A, B and C for H

sat 2 R134A using tabulated data for T O and vs. P.

Compare T sat at P = 0.25P c and 0.7P c obtained

from the relation with the tabulated values.

Problem G20 A

Determine the chemical potential of liquid CO 2 at 25ºC and 60 bar. The chemical poten- tial of CO 2 , if treated as an ideal gas, at those

conditions is –451,798 kJ kmol . Problem G21

Plot P(v) in case of H 2 O at 373 K in the range v min = 0.8*v f and v max = 1.5*v g assuming that

the fluid follows the RK state equation. The values of v f and v

are (for 523 K, P g sat )

exp(.582(1-T sat c /T)). What are the values for v –1 and v f

g for P ?. Assume that h = 0 kJ kmol

Problem Figure G.15 K. From the g(P) plot, determine the RK satu-

and s = 0 kJ kmol –1 K –1 at v = 0.8v f and 523

ration pressure at 523 K. Problem G22

The properties of refrigerant R–134A (CF 3 CH 2 F) are required. The critical properties

c = 512.2 kg/m , M = 102.03, h fg = 217.8 kJ kg –1 ,T freeze = 172 K, and T NB = 246.5 K (the normal boiling point is the satu- ration temperature at 100 kPa). Plot the values of ln (P sat ) with respect to 1/T using

of the fluid are T c = 374.2 K, P c = 4067 kPa, ρ

Clausius–Clapeyron equation. Use the RK equation of state and plot P R with respect to V R with T R as a parameter. Use the relation dg T = vdP = ( ∫d(pv) – ∫Pdv) to plot the

values of g/RT c with respect to v ′ R at specified values of T R . Assume that g/RT c = 0 at 373 K when v ′ R = 0.1.

Problem G23 You are asked to analyze the internal energy of photons which carry the radiation en- ergy leaving the sun. Derive an expression for change in the internal energy of the photons if they undergo isothermal compression from a negligible volume to a vol-

ume v. The photons behave according to the state equation P = (4 σ/3 c )T 4 0 , where σ = 5.67 ×10 –11

0 ×10 10 kW m –1 K =3 ms the speed of light in vacuum, and T the temperature of the radiating sun.

–4 denotes the Stefan Boltzmann constant, c

a) Show that c v =c v (T, v) for the photons. b)

Obtain a relation for µ. Problem G24

From the relation s = s(T,P), obtain a relation for ( ∂T/∂P) s in terms of c p , β P , v and T. If Z = 1 + ( αT R + βT m R )P R , where α = 0.083, β = –0.422, and m = 0.6, obtain an ex- pression for (s o – s)/R.

Problem G25 How much liquid can you form by throttling CO 2 gas that is at 200 bar and 400 K to 1 bar? The property tables are not available. How much liquid can you form if you use an isentropic turbine to expand the gas to 1 bar? Make reasonable assumptions.

Problem G26 Recall that du = c v dT + (T( ∂P/∂T) v – P) dv. A) Obtain an expression for du for a VW gas. Is c v

a function of volume? (Hint: use the Maxwell’s relations.) B) If c vo is inde- pendent of temperature, obtain an expression for the internal energy change when the temperature and volume change from T 1 to T 2 and from v 1 to v 2 . Assume c v is con- stant.

Problem G27 Gaseous N 2 is stored at high pressure (115 bar and 300 K) in compartment A (that has

a volume V A ) of a rigid adiabatic container. The other compartment B (of volume V B = 3V A ) contains a vacuum. The partition between them is suddenly ruptured. If c v =

c vo = 12.5 kJ kmol –1 K –1 , determine the temperature after the rupture. Assume VW gas.

Problem G28 Gas from a compressed line is used to refill a gas cylinder from the state (P 1 ,T 1 ) to a pressure P 2 . The line pressure and temperature are P i and T i . Determine the final pres- sure and temperature if (a) the cylinder is rapidly filled (i.e. adiabatic) and (b) slowly filled (i.e. isothermal cylinder). Use the real gas state equation P = RT/(v–b) – a/v 2 .

Problem G29

∆h vapor ∆v vapor Using the relation ln P sat = (A – B/T), show that = (BP /T). Problem G30

sat

Derive an expression for (a(T,v)–a o (T,v o )) using the Peng–Robinson equation P = (RT/(v–b)) – a(w,T)/(v(v+b) + b(v–b)). Derive expressions for (s–s o ) and (h–h o ).

Problem G31

a) Calculate the fugacity of H 2 O(1) at 400 psia and 300 °F (assume that v = c for the liquid state). b) If the condition A denotes compressed liquid, then is f A (P,T) ≈f(T,P sat ) (i.e., the fugacity of the saturated liquid at the same temperature)? c) At phase equilib-

rium the fugacity of the saturated liquid equals that of saturated vapor, i.e., f(T, v f )= f(T, v

g ) and P(T, v f ) = P(T, v g ). Predict P at 200 °C using these relations and the RK state equation.

sat

Problem G32 For a Clausius gas, P(V – Nb) = NRT, and for a Van der Waals gas P = NRT/(V –

2 Nb) – N 2 a/V . For either gas obtain expressions for ( ∂P/∂T)

V ,( ∂P/∂v) T , and ( ∂v/∂T) P . If for a pure substance, ds = ( ∂P/∂T) v dv + (c v /T)dT, show that for both gases c v is in- dependent of the volume.

Problem G33 The differential entropy change for a gas obeying the molar equation of state p = RT/v – aT 2 /v is ds = (A/T – 2a ln v) dT + (R/v – 2aT/v) dv. Perform the line integra- tion from state (v 1 ,T 1 ) to (v 2 ,T 2 ) along the paths (v 1 ,T 1 ) → (v 2 ,T 1 ) → (v 2 ,T 2 ), and

(v 1 ,T 1 ) → (v 1 ,T 2 ) → (v 2 ,T 2 ) and show that (TdS – Pdv) is exact. Problem G34

In the section of the liquid–vapor equilibrium region well below the critical point v l «v g and the ideal gas law is applicable for the vapour. Derive a simplified Clapeyron equation using these assumptions and show how the mean heat of vaporization can be

determined if the vapor pressures of the liquid at two specified adjacent temperatures are known.

Problem G35 For ice and water c = 9.0 and 1.008cal K –1 mole –1 , respectively, and the heat of fu- sion is 79.8 cal g –1 p at 0ºC. Determine the entropy change accompanying the spontane- ous solidification of supercooled water at –10ºC and 1 atm.

Problem G36 For water at 110ºC, dP/dT = 36.14 (mm hg) K –1 and the orthobaric specific volumes are 1209 (for vapor) and 1.05 (for liquid) cc g –1 . Calculate the heat of vaporization of water at this temperature.

Problem G37 The specific heat of water vapor in the temperature range 100º–120ºC is 0.479 cal g –1

K –1 , and for liquid water it is 1.009 cal g –1 K –1 . The heat vaporization of water is 539 cal g –1 at 100ºC. Determine an approximate value for h fg at 110ºC, and compare this result with that obtained in the previous problem.

Problem G38 Recall that dg T = v dP, and plot g ′ R (= (g/RT c )) and P R with respect to v ′ R at 593 K

for H 2 O and determine the liquid like and vapor like solutions at 113 bar. Determine saturation pressure at T = 593 K for RK fluid. Assume that g = 0 at v ′ R = 200.

Problem G39 Use the expression du = c v dT + (T( ∂P/∂T) v – P) dv to determine c v for N 2 at 300 K

and 1 bar. Integrate the relation along constant pressure from 0 to 300 K at 1 bar, and then from 1 to 100 bar at 300 K in the context of the RK equation. What is the value of u at 300 K and 100 bar if u(0 K, 1 bar) = 0?

Problem G40 Since T = T(S,V,N) is an intensive property, it is a homogeneous function of degree zero. Use the Euler equation and a suitable Maxwell relation to show that ( ∂T/∂v) s =

–sT/c v v, and ( ∂P/∂s) v = sT/(c v v). For a substance that follows an isentropic process with constant specific heats, show that T/v (s/cv) = constant

Problem G41 Show that generally real gases deliver a smaller amount of work as compared to an ideal gas during isothermal expansion for a (a) closed system from volume v 1 to v 2 (Hint: use the VW equation ignoring body volume), and (b) an open system from pressure P 1 to P 2 (Hint: use the fugacity charts in the lower pressure range).

Problem G42 Plot the values of (c v –c vo ) with respect to volume at the critical temperature using the RK state equation. What is the value at the critical point?

Problem G43 Assume that the Clausius Clapeyron relation for vapor–liquid equilibrium is valid up to the critical point. Show that the Pitzer factor w =0.1861 (h fg /RTc)-1. Determine the

Pitzer factor of H –1

2 O if h fg = 2500 kJ kg .

Problem G44 An electron gas follows the relation S = C N 1/6 V 1/3 U 1/2 . Obtain an expression for c v and show that c v =c v (T,v). Also obtain expressions for u(T,v) and h(T,v).

Problem G45 Determine the values of u, h and s at 444 K and 1000 kPa for Freon 22, (Chlorodifluromethane) if s = 105.05 kJ kmol –1 K –1 o , h o = 32667 kJ kmol –1 , and M =

86.47 kg kmol –1 . Use the RK equation.

Problem G46 Upon the application of a force F a solid stretches adiabatically and its volume in- creases by an amount dV. The state equation for the solid is P = BT m (V/V o – 1) n . Show that the solid can be either cooled or heated depending upon the value of m.

Problem G47 Use the Peng-Robinson equation to determine values of P sat (T) for H 2 O.

Problem G48 Apply the Clausius Clapeyron equation in case of refrigerant R–134A. Assume that

h fg, = 214.73 kJ kg –1 , at T ref = 247.2 K, and P ref = 1 bar. Discuss your results, and the

impact of varying h fg .

Problem G49

A superheated vapor undergoes isentropic expansion from state (P 1 ,T 1 ) to (P 2 ,T 2 ) in a turbine. It is important to determine when condensation begins. Assume that vapor behaves as an ideal gas with constant specific heats. Assume that ln P sat (in units of bar) = A – B/T(in units of K) where for water A = 13.09, B = 4879, and c vo = 1.67 kJ kg –1 . a)

Obtain an expression for the pressure ratio P 1 /P 2 that will cause the vapor to

condense at P 2 .

b) Qualitatively sketch the processes on a P-T diagram. Problem G50

Determine the chemical potential of CO 2 at 34 bar and 320 K assuming real gas be- havior, h –1 o =c po (T – 273), s o =c po ln (T/273) – R ln (P/1), and c po = 10.08 kJ kmol .

Problem G51 Does H 2 O(g) (for which P c = 221 bar and T c = 647 K) behave as an ideal gas at 373 K

and 1.014 bar? Determine the value of v g .

Problem G52 What is the enthalpy of vaporization h

fg of water at 373 K if P = 1.014 bar? Assume

sat

RK equation and v = 0.001 m 3 f / kg.

Problem G53 R134A is stored in a 200 ml adiabatic container at 5 bar and 300 K. It is released over a period of 23 ms during which the mass decreased by 0.32 g. Assume that R134A behaves according to the RK state equation and its ideal gas specific heats are not functions of temperature. Obtain a relation between temperature and volume for the isentropic process in the tank. Using this relation, determine the pressure and tem- perature in the tank after the R134A release. If the process were isentropic with con- stant specific heats, what would be the pressure and temperature in the tank after the release of R134A?

Problem G54 Determine the closed system absolute availability φ of a fluid that behaves according

to the RK equation of state as it is compressed from a large volume v 0 at a specified temperature. Assume that u = 0, s =0, and φ = 0 at the initial condition. Obtain an ex-

pression for f(v, T, a, b). (Hint: first obtain expressions for u and s.) Determine φ for

H 3 2 –1 O at 593 K and a specific volume of 0.1 m kmol . Use v 1 at 1 bar and 593 K. Problem G55

Using the result (c p –c v ) = T( ∂v/∂T) P ( ∂P/∂T) v show that if Pv = ZRT, then(c p –c v /R) = Z + T R (( ∂Z/∂T R ) v ′ R +( ∂Z/∂T R )

2 P R ) + (T R /Z)( ∂Z/∂T R ) v ′ R ( ∂Z/∂T R ) P R . Can you use the

“Z charts” for determining values of (c p – c v ) for any real gas at specified tempera- tures and pressures?

Problem G56 It is possible to show that (c p –c v )=vT β 2 P / β T , and, for VW gases, c v =c vo . For a VW gas show that (c p –c v ) = c p (T,v) – c

vo (T) = R/(1 – (2a(v–b) )/(RTv )). Determine the value of c p at 250 bar and 873 K for H 2 O if it is known that c vo (873 K) = 1.734 kJ kg –1

K. Compare your results with the steam tables. Problem G57

If (b/v) 2 « (b/v) in context of the state equation P = RT/(v–b) – a/ T n 2 v , an approxi- mate explicit solution for v(P,T,a) is v = α + (–β + α 2 ) 1/2 = α (1 (1–β/α 2 ) 1/2 ), β/α 2 <1,

where α(T,P)= RT n+1 /(2PT n ), and β(T,P)= (a–bRT n+1 )/(PT n ). If h = u o – a/v + Pv, ob- tain an expression for c p .

Problem G58 Use the RK state equation to plot (g–g ref )/RT c with respect to P R for values of T R = 0.1,0.2, ..., 1.0. Also plot P sat R vs. T R .

Problem G59 Develop a computer program that calculates P sat R with respect to T R using the RK

equation of state and the criterion that g f =g g .

Problem G60 Obtain values of T inv,R with respect to v ′ R , and T inv,R and Z inv with respect to P inv,R us-

ing the RK equation of state. Problem G61

In the context of throttling, cooling occurs only if the temperature T<100 K for H 2 and T<20 K for He. Check this assertion with the expression for µ JT based on VW

JT = – (1/c p ) (v – ((RT/(v–b))/(RT/(v–b) – 2a/v )) for both fluids. Problem G62

state equation µ

A rigid adiabatic container of volume V is divided into two sections A and B. Section

A consists of a fluid at the state (P A,0 ,T A,0 ) while section B contains a vacuum. The partition separating the two sections is suddenly ruptured. Obtain a relation for the change in fluid temperature with respect to volume (dT/dv) after partition is removed in terms of β P , β T , P, and c v . What is the temperature change if the fluid is incom-

pressible? What is the temperature change in case of water if V A = 0.99 V, P = 60 bar, β

3 and T = 30ºC, –1

bar ,v A = 0.00101 m kg , and c p =c

P = 2.6

K , T = 44.8

v = c = 4.178 kJ kg K? Problem G63

Trouton’s empirical rule suggests that ∆s fg ≈ 88 kJ kmol –1 K –1 at 1 bar for many liq- uids liquids (another form is h fg = 9 RT NB ). Obtain a general expression from the Clau-

sius Clapeyron equation for the variation of saturation temperature with pressure.

Problem G64 Using the state equation P = RT/(v–b) – a/(T n v m ) and the equality g f = g g , show that P sat = (1/(v

g –v f )) (RT ln ((v g –b)/(v f –b)) + (a/(m–1)T ) (1/v g – 1/v f )). Simplify the result for the VW and Berthelot equations of state.

(m–1) (m–1)

Problem G65

32 Show that the inversion temperature for an RK gas / T

= (4.9342(1 – 008664/ 2 v ′ R ) /(1+ 0.08664/ v ′ R )) (1.5 + 1/(1+0.08664/ v ′ R )).

R inv ,

Problem G66 Using the relations ds = c v dT/T + ∂P/∂T dv and ds = c p dT/T – ∂v/∂T dP, show that

(c

P –c v )=Tv β P / β T .

Problem G67 The Helmholtz function A for a Debye solid A = –N R π 4 T 4 /(5 θ 3 D ), where θ D is De-

bye temperature and N denotes the number of moles. Obtain expressions for u, s, and

c v , in terms of the temperature. Problem G68

Determine the relation between the temperature and volume during an isentropic process for a VW gas. If at the initial state 1, T 1 = 200 K, v 1 = 0.006 m 3 kg –1 , and if

v 1 /v 2 = 3, determine the final state 2 (P 2 ,T 2 ) if the gas is air.

Problem G69 In the context of the relation s = s(u,T) show that P/T is only a function of volume as v → ∞ for any simple compressible substance.

Problem G70 About 0.1 kmol of liquid methanol at 50ºC in system A is separated by a thin foil in thermal and mechanical equilibrium from dry N 2 occupying 1 % of liquid volume at 2 bar and 50ºC in system B. The foil is removed and the liquid temperature falls. Heat must be consequently added to maintain the state at 50ºC and 2 bar in both subsys- tems. Determine the partial pressure of vapor at which the vaporization stops. Assume

that h fg = 37920 kJ kmol –1 . If µ methanol (l) = g methanol = h(l) – T s(l), µ methanol (g) = g metha- nol (g) = h(g) – T s(T,p methanol ), and p methanol =X methanol P. Neglect the volume change in

the liquid methanol. Determine G = G A +G B with respect to p methanol . Problem G71

Show that the chemical potential of a pure VW gas is µ(T,v) = µ(T,P) = µ o (T) + RTv/(v–b) – 2a/v – RT – RT ln (pv/RT) + RT ln (v/(v–b)).

Problem G72

Apply the Martin–Hou state equation P = RT/(v–b) + (B v) Σ i=2,5 F i (T)/(v–b) +F 6 (T)/e , for which b and B are constants to obtain expressions for a(T,v)–a 0 (T,v), s(T,v)

–s 0 (T,v), and u(T,v)–u 0 (T). Let dF(T)/dT = F´(T). What are the expressions for the case if F –KT/Tc

i =A i +B i T+C i e ? (ASHRAE tabulates these constants for various re- frigerants.)

Problem G73 Determine the temperature after C H is throttled from 20 bar and 400 K 1 bar with

p.o = 94.074 kJ kmol K . Use a) RK equation and b) Kessler charts for h /RT c . Problem G74

c –1 –1

Consider du = c v dT + (T( ∂P/∂T) v – P) dv. Obtain a relation for u 0 -u and c v0 -c v in terms of a,b, n,T and v for generalized RK equation of state P = RT/(v-b)- a/(T n v

(v+b)).

Problem G75 Using RK equation of state and appropriate reference conditions determine the fol- lowing for steam at 180 bar and 400ºC and compare the values with steam tables for v, h,u and s and fugacity charts for f/P: v, h, u, s, c v ,c p -c v ,c p , f/P and µ JT in K /bar.

Problem G76 The following expression for the Helmholtz function has been used to determine the properties of water

a( ,T) = a (T) + RT[ ρ o ln ρ + Q( , )] ρρτ , where ρ denotes density, T denotes temperature on the Kelvin scale, τ denotes

1000/T. The functions a 0 and Q are sums involving the indicated independent vari- ables and a number of adjustable constants, i.e.,

∑ i-1 C/ τ + C 7 ln T + C 8 ln T/ τ , and

i-1

Q = (-) ττ c ∑ (- ττ aj )  ∑ A(- ij ρρ

j-2 

aj ) + e ∑ A ij ρ .

 Here, R = 4.6151 bar cm 3 /g K or 0.46151 J/g . K, τ c /1000/T c = 1.544912, E = 4.8, and

τ aj = τ c if j=1, τ aj = 2.5 if j>i, ρ aj = 0.634 if j=1, ρ aj =1.0 if j>i.

The coefficients for a 0 in joules per gram are given as follows;

C 5 = -20.5516

C 8 = -1011.249

C 3 = -419.465

C 6 = 4.85233

Values for the coefficients A aj are listed in the original source. Obtain expressions for (a) pressure, (b) specific entropy, (c) specific internal energy and specific enthalpy re- sulting from this fundamental function. See also J.H. Keenan, F.G. Keyes, P.G. Hill, and J.G. Moore, Steam Tables, Wiley, New York, 1969; L. Haar, J.S. Gallagher, and G.S. Kell, NBS/NRC Steam Tables, Hemisphere, Washington, D.C., 1984. The properties of water are determined in this reference using a different functional form for the Helmholtz function than given by Eqs. (1)-(3).

Problem G77 Ammonia is throttled from P 1 =169 bar and T 1 = 214 C to a very low pressure P 2 (<< critical pressure). Determine a)

T 2 in C and b)

Change in internal energy u 2 -u 1 in kJ/kg

Use Kessler tables and ignore Pitzer factor. The ideal gas specific heat can be as- sumed to be a constant and equal to c p0 = 2.130 kJ/kg K, M= 17.03 kg/kmol.