CHAPTER 6 PROBLEMS

F. CHAPTER 6 PROBLEMS

Problem F1 Can the combustion gases emerging from a boiler be considered to have the same properties as air or should we employ the real gas equation of state?

Problem F2 What is v c ´?

Problem F3 Which two–parameter equation of state is best to use?

Problem F4 Which two–parameter equation of state does not yield negative pressures?

Problem F5 What are the important differences between the Dietrici and VW equations of state?

Problem F6 Why do we obtain two solutions when we neglect the parameter b in the VW equation of state?

Problem F7 Is there an analytical method for determining the stability of solutions?

Problem F8 Are there generalized equations of state for complex fluids that do more than just Pdv (i.e., compressible) work?

Problem F9 Is the real gas equation of state valid for high speed flows as long as they are in con- tinuum?

Problem F10 Does v → 0 as P → ∞, T → 0?

Problem F11 Is it true that at specified values of T R and Z, P R is single valued?

Problem F12 Is it possible to develop a real gas equation of state for a subcooled liquid?

Problem F13 Is the Pitzer factor constant for any given fluid?

Problem F14 Why do equations of state sometimes fail in the compressed liquid and vapor domes?

Problem F15 Why is the vapor dome region difficult to predict with a two–parameter equation of

Problem F16 Can we extend the real gas equation of state to liquids?

Problem F17 Can you determine the value of Z with just the values of T R and P R for the Clausius II equation of state?

Problem F18 What are the values of Z c for the RK, VW, Berthelot, and Dietrici equations of state?

Problem F19

Is it true that real gas equations of state are applicable only for the vapor state?

Problem F20 How are real gas state equations derived?

Problem F21 Why do some state equations predict saturated properties well, while others do not?

Problem F22 Why is the “middle” solution for v at specified values of T and P meaningless in context of a cubic equation?

Problem F23 How many distinct real solutions exist in context of the RK equation at a specified

temperature if T > T c , and T < T c ?

Problem F24 The fundamental equation for an electron gas is S = C N 1/6 (V 2/3 U – (3/5)C N 5/3 1 2 ) 1/2 . Obtain an equation of state in terms of P, V and T. Does this electron gas behave as an ideal gas? What is the compressibility factor at 200 bar and 100 K?

Problem F25 Consider the VW equation P = RT/(v–b) – a/v 2 . Plot the P(v) behavior of water. Show that P 2

R =T R /( v ′ R –1/8) – (27/64)/ v ′ R and plot P R with respect to v ′ R for T R = 0.6 and

1.2. Prove that the Z > 1 when the body volume effect dominates attractive forces (i.e., a ≈0 at very high pressures) and vice versa (i.e., b/v «1, (b/v) 2 ≈0, but b≠0). Us-

ing the relation, Z = P R v ′ R /T R plot Z(P R ) for T R = 0.6 and 1.2 and Z(P R ) for v R ′ =0.3 and 0.4, and discuss your results. Problem F26 Derive an expression for a and b in terms of T c , and P c for the Dietrici equation of

state P = (RT/(v–b)) exp(–a/(RTv)) and show that a = (4/P )(RT

c c /e) 2 and b = RT c /(P c

e ) where e = 2.7182818. Note that one cannot obtain negative pressures with the Dietrici equation as opposed to the RK equation unless v « b, which is physically im- possible. Plot P(v) for water at various temperatures and obtain gas like solutions for volume vs. T (if they exist) at 113 bar.

Problem F27 For the Clausius II equation, obtain the relations for a, b, and c in terms of critical properties and critical compressibility factor. (Hint: Solve for a and b in terms of c

and v c using the inflection condition. Then, use the tabulated value of Z c to determine that of c.) Determine the corresponding values for H 2 O and CH 4 .

Problem F28 Calculate the specific volume of H 2 O(g) at 20 MPa. and 673 K by employing the (a) compressibility chart, (b) Van der Waals equation, (c) ideal gas law, (d) tables, (e)

Pitzer correction factor and Kessler tables. What is the mass required to fill a 0.5 m 3 cylinder as per the five methods?

Problem F29 Determine the specific volume and mass of CH 4 contained in a 0.5 m 3 cylinder at 10 MPa and 450 K using the following methods: a)

Ideal gas law. b)

Compressibility charts. b)

van der Waals equation. c)

Approximate virial equation of state. d)

Compressibility factor tables including the Pitzer factor. e)

Approximate equation for v(P,T) given by expanding the Berthelot equation v = (1/2)(b +(RT/P))(1±(1–(4a/(PT(b+RT/P)))) 1/2 ), b/v «1.

Problem F30 Consider the virial equation of state (Pv/RT) = Z = 1 + B(T)/v + C(T)/v 2 . a)

Determine B(T) and C(T) if P = RT/(v–b) and b/v «1. b)

Determine B(T) and C(T) if P = RT/(v–b) – a/v 2 and b/v « 1. i)

Obtain an expression for the two solutions for v(T,P) from the quadratic equation. Are these solutions for the liquid and vapor states? Discuss.

ii) Discuss the two solutions for steam at 373 K and 100 kPa. Explain the significance of these solutions.

iii) Show that the expression for the Boyle temperature (at which Z = 1) is provided by the following relation if second order effects are ig- nored, namely, T Boyle = a/(Rb).

iv)

What is the Boyle temperature for water?

Problem F31 CF 3 CH 2 F (R134A) is a refrigerant. Determine the properties (v, u, h, s, etc.) of its va- por and liquid states. The critical properties of the substance are T c = 374.2 K, P c = 4067 kPa, ρ c = 512.2 kg m –3 , M = 102.03 kg kmol –1 ,h fg = 217.8 kJ kg –1 ,T freeze = 172

K, T NB = 246.5 K (this is the normal boiling point, i.e., the saturation temperature at 100 kPa). a)

Determine the value of vs at (liquid) at 247 K. Compare your answer with tabulated values (e.g., in the ASHRAE handbook).

b) Determine the density of the compressed liquid at 247 K and 10 bar. c)

Use the RK equation to determine the liquid and vapor like densities at 247 K and 1 bar. Compare the liquid density with the answer to part (b).

Problem F32

2 If c = –kv 2 (dP/dv) T , deduce the relation for the sound speed of a RK gas in terms of

v ′ R ,T R , and k. Problem F33

Using steam tables, determine β P and β T for liquid water at (25ºC, 0.1 MPa)., (70ºC,

0.1 MPa), and (70ºC, 10 MPa). What is your conclusion? Problem F34

Show that if (b/v) 2 «(b/v), the explicit solutions for v(P,T) and a in the context of the state equation P = RT/(v–b)–a/T n v 2 are provided by the relations v = α(1+(1–β/α 2 ) 1/2 ),

β/α 2 <1, where α(T,P) = RT n+1 /(2PT n ), β(T,P) = (a – bRT n+1 )/(PT n ). (Hint: expand the term 1/(v–b) as a polynomial in terms of (b/v).) Using the explicit solutions with n = 0

(i.e., the VW state equation), determine the solution(s) for v(593 K, 113 bar) in the (i.e., the VW state equation), determine the solution(s) for v(593 K, 113 bar) in the

Problem F35

A diesel engine has a low compression ratio of 6. Fuel is injected after the adiabatic reversible compression of air from 1 bar and 300 K (state 1) to the engine pressure (state 2). Assume that for diesel fuel P c = 17.9 bar, T c = 659 K, ρ 1 = 750 kg m –3 ,C p1 =

2.1 kJ kg –1 K –1 , ∆h c = 44500 kJ kg –1 ,L 298 = 360 kJ kg –1 , L(T) = L 298 ((T c – T)/(T c –

298)) sat , and log

10 P = a – b/(T – c), where a = 4.12, b = 1626 K, c = 93 K. Deter- mine the specific volume of the liquid at 1 bar and 300 K. Assume that the value of Z c can be provided by the RK equation. Since the liquid volume does not significantly change with pressure, using the value of the specific volume and ρ l determine the fuel

0.38 sat

molecular weight. Determine the liquid specific volume at state 2. What are the spe- cific volumes of the liquid fuel and its vapor at the state (P sat ,T

2 )? Problem F36

Derive an expression for f/P for VW gas using the definition dg = RT d ln (f) = v dP and dg ig = RT d ln (P) = v ig dP; determine f/P at critical point using the expressions of “a” and “b” for VW gas.

Problem F37 Determine the values of v l and v g for refrigerant R–12 at 353 K and 16 bar by apply- ing the following models: a) ideal gas, b) RK equation, c) PR equation, d) Rackett

equation, e) PR equation with w = 0. Discuss the results. Problem F38

Experimental data for a new refrigerant are given as follows:

P 1 = 111 bar,T 1 = 365 K, v 1 = 0.1734m 3 /kmol P 2 = 81.29 bar, T 2 =T 1 = 365, v 2 = 0.2805

a) If VW equation of state is valid, determine “a” and “b”

b) If critical properties P c , T c of the fluid are not known, how will you deter- mine T c ,P c ? Complete solution is not required.

Problem F39

3 The VW equation of state can be expressed in the form Z 2 – (P

R /(8T R ) +1)Z + (27 P

R /(512 T R ) )= 0. Obtain an expression for ∂Z/∂P R and its value as P R → 0. At what value of T R is ∂Z/∂P R =0. Obtain an expression for an approxi- mate virial equation for Z at low pressures. Problem F40

R /(64T R )) Z – (27 P

For the Peng–Robinson equation of state: a = 0.4572 R 2 T 2 c /P c and b = 0.07780 R T c /P c . Determine the value of Z c , and Z(673 K, 140 bar) for H 2 O.

Problem F41

Consider * the state equation: P

R =T R /( v ′ R –b )–a (1+ κ(1– T 1/2 R )) 2 /(T n R (( v ′ R +c * )+(v R '+d * ))), where n = 0 or 0.5, and κ is a function of w only. If P R (( v ′ R +c * )+( v ′ R +d * ))/a * = A, and (( v ′ R +c * )+( v

′ * R +d * ))/(a * ( v ′ R –b )) = B, show that for n = 0, P R =T R /( –b * )–a * (1+ κ(1–T 1/2 )) 2 / (T n ((

v 1/2

v ′ R +c )+( v R ′ +d ))), and T R = –( κ+κ 2 )/(B– κ 2 ) ± ((1+2 κ +κ 2 κ 2 κ+κ 2 + A)/(B– 2 )+( ) /(B– κ 2 2 ) 1/2 ) .

Problem F42 Consider the state equation P R =T R /( v ′ R –b * ) – a * /(T n R (( v ′ R +c * ) v ′ R ))). Show that for the Berthelot

and Clausius II equations (n

= 1), T =

P R ( v ′ R –b )/2(1+(4a ( v R ′ –b )/( v ′ R +c ) +1) ). Show that for the VW equation of state, both n and c * equal zero, that c * =d * = 0 for the Berthelot equation, and d * = 0 for the Clausius II equation.

Problem F43 Plot the pressure with respect to the specific volume of H 2 O by employing the RK state equation at 600 K and determine the liquid– and vapor–like solutions at 113 bar.

Problem F44 Plot the product P v with respect to the pressure for water (you may use tabulated values). Does low pressure P v provide any insight into the temperature. Can you construct a constant volume ideal gas thermometer which measures the pressure in a glass bulb containing a known gas and then infer the temperature?

Problem F45 Using the inflection conditions for the Redlich–Kwong equation P = (RT/(v–b)) – a/(T 1/2 v(v+b)), derive expressions for a and b in terms of T c , and P

3 2 c (b/v . and show that (a)

c ) – 3(b/v c ) – 3(b/v

c ) + 2 = 0, or b/v c = 0.25992, (b) a/ (RT c v c )=

c )), or a/ (RT c v c )= 1.28244, and (c) Z c = 1/3. Problem F46

(1+(b/v c ) 2 )/((1– (b/v c ) 2 ) (2 + (b/v

Determine explicit solutions for v(P,T) if (b/v) 2 «< (b/v) for the state equation P = RT/(v–b) – a/(T n v(v+b)). Show that 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 ), β(T,P)= (a – bRT n+1 )/(PT n ). (Hint: expand 1/(v–b) and 1/(v+b) in terms of polynomials of (b/v).) Using the explicit solutions and n = 1/2

(RK equation), determine solutions for v(593 K, 113 bar) for H 2 O. Show that if v » b n then Z < 1, and if RT/(v–b)»a/T 2 v (i.e., v ≈ b, or that the molecules are closely

packed) then Z > 1. Problem F47

Using the RK equation obtain an approximate expression for v by neglecting terms of

the order of (b/v) 3 .

Problem F48 Convert the Berthelot, VW, and Dietrici state equations to their reduced forms using

the relations P R = P/P c ,T R = T/T c , and v ′ R = v/ v ′ c , v ′ c = RT c /P c .

Problem F49

For the state equation P = RT/(v–b) – a/(T 2 v ) show that (a) a = ((m+1) /4m) (RT n+1 m–1 c v c ), b = v c (1– (2/(m+1))), and Z c = ((m 2 –1)/(4m)). Obtain a reduced form

of this real gas equation, i.e., P R = f( v ′ R ,T R ). Problem F50

For the state equation P = RT/(v–b) – a/ v 2 determine the values of a and b without using the inflection conditions, but using the facts that at critical point there are three equal real roots (at T<T c there are three roots, and for T > T c only one real root ex- ists).

Problem F51 Determine the Boyle curves for T R vs. P R for gases following the VW equation of state. Also obtain a relationship for P R ( v ′ R ).

Problem F52 If number of molecules per unit volume n´ = 1/l 3 where l denotes the average distance

(or mean free path between molecules). determine the value of l for N 2 contained in a cylinder at –50ºC and 150 bar by applying the (a) ideal gas law and (b) the RK equa-

tion. Compare the answer from part (b) with the molecular diameter determined from the value of b . Apply the LJ potential function concept (Chapter 1) in order to deter-

mine the ratio of the attractive force to the maximum attractive force possible.

Problem F53 In the case of the previous problem determine the value of l for the H 2 O at 360ºC and 120 bar.

Problem F54 Using the RK equation plot the pressure with respect to specific volume at the critical temperature for the range 0.25v c <v<2v c . Here, v c has its value based on the RK equa- tion at specified values of P c and T c . From the tables plot the function P(v) for the same conditions and discuss your results.

Problem F55 Apply the RK equation for H 2 O at 473 K, 573 K, and 593 K and obtain gas–like so- lutions (if they exist) at 113 bar. Compare these values with the liquid/vapor volumes obtained from the corresponding tables.

Problem F56

A person thinks that the higher the intermolecular attractive forces, the larger the amount of energy or the higher the temperature required to boil a fluid at a specified pressure. Consequently, since the term a in the real gas equation of state is a measure of the intermolecular attractive forces, you are asked to plot T sat with respect to a. Use the normal boiling points (i.e., T sat at 1 bar) for monatomic gases such as Ar, Kr, Xe, He, and Ne, and diatomic gases such as O ,N , Cl , Br ,H , CO, and CH . Also de- termine T sat

2 2 2 sat 2 2 4

using the correlation ln(P R ) = 5.3(1–(1/T R )) where P R = P/P c and P = 1 bar. Use the RK and VW state equations. Do you believe the hypothesis?

Problem F57

A fixed mass of fluid performs reversible work δW = Pdv according to the processes 1–2 isometric compression, 2–3 isothermal heating at T H , 3–3 isometric expansion,

and 4–1 isothermal cooling at T L . The cycle can be represented by a rectangle on a T–v diagram. Determine the value of ∫δW/T if the medium follows the VW and ideal gas equations of state.

Problem F58 Flammable methane is used to fill a gas cylinder of volume V from a high–pressure compressed line. Assume that the initial pressure P 1 in the gas tank is low and that the temperature T 1 is room temperature. The line pressure and temperature are P i and T i . Typically, P i »P 2 , the final pressure. There is concern regarding the rise in temperature

during the filling process. We require a relation for T 2 and the final mass at a speci- fied value of P 2 . Assume two models: (a) the ideal gas equation of state P = RT/v for which du

0 =c vo dT, and (b) the real gas state equation P = RT/(v–b) – a/v with c v =

c vo and du = c v dT +(T ∂P/∂T – P)dv. Problem F59

Determine v for water at P =133 bar, T= 593 K using VW, RK, Berthelot, Clausius II, SRK and PR equations.

Problem F60 Consider generalized equation of state P = RT/(v-b) - a α (w, TR) / (T n (v+c) (v+d)). Using the results in text, determine Z and v for H2O atT1 = 473K, P1 = 150 bar, T2=

873K, P2 = 250 bar using VW, RK, Berthelot, Clausius II, SRK and PR. Compare re- sults with steam tables.