Cxhapter 10

10.A-1 One of the first advanced equations was the Beattie-Bridgeman equation of state originally proposed in 1928 which has the following form.

30

02 2

11 1

cRTv T Ab aP v B

v v v v

A0, B0, a, b, and c, are substance-dependent coefficients. Values of these coefficients for many gases can found in the literature. For carbon dioxide, the gas of interest in this problem, the coefficients have the values provided in Table 10.A-1.

Table 10.A-1: Coefficients for the Beattie-Bridgeman equation of state for CO2 (from Cravalo and Smith, Jr., Engineering Thermodynamics, Pitman, Boston, 1981

a 1.62129e3 [m3/kg] A0 262.07 [N-m4/kg2] b 1.6444e3 [m3/kg] B0 2.3811e3 [m3/kg] c 1.4997e4 [m3-K3/kg]

a.) Test the Beattie-Bridgeman equation's ability to accurately predict the specific volume of carbon dioxide by comparing the specific volumes and compressibility factors obtained from the equation of state with a reliable source for isotherms at 250 K, 304.1 K and 350 K and pressures ranging from atmospheric to 200 bar. What is your assessment of the accuracy of the Beattie-Bridgeman equation of state for carbon dioxide?

b.) It is difficult to accurately measure the critical volume, so there is some incentive to obtain it from the equation of state. One way to do so is to plot the slope dP/dv along the critical isotherm to determine the volume at which it is zero. Can you obtain an accurate estimate of the critical volume from the Beattie-Bridgeman equation of state?

S.A. Klein and G.F. Nellis Cambridge University Press, 2011

10.A-2 The Benedict-Webb-Rubin (BWR) equation of state, originally proposed in 1940, has the following form.

2

2 3 6 3 2 2 2

/ 1 expo o oB RT A C TRT b RT a a cPv v v v v T v v

Ao, Bo, Co, a, b, c, and are substance-dependent coefficients. Values of these coefficients can be found in the literature. For methane, the gas of interest in this problem, the coefficients have the values indicated in Table 10.A-2 with P in Pa, v in m3/kmol and T in K. Table 10.A-2: Coefficients for the Benedict-Webb-Rubin equation of state for methane (Van Wylen, G. and Sonntag, R., Fundamentals of Classical Thermodynamics, 3rd edition, Wiley, New York, 1986)

a = 5000 Ao = 187.91E3 b = 0.003380 Bo = 0.04260 c = 2.578e8 Co = 2.286e9 =1.244e-4 = 0.0060

a.) Determine the units of the eight coefficients b.) Test the BWR equation's ability to accurately predict the specific volume of methane

in the superheated, subcooled and saturated regimes by comparing the specific volumes obtained from the equation with a reliable source. Summarize your results in a table and indicate the percentage error.

c.) It is much more difficult to accurately measure the critical volume than it is to measure critical temperature and pressure. Assuming that the BWR constants provided above are correct, determine the critical volume at which the critical isotherm exhibits a slope of 0. How does your result compare to the accepted value?


10.A-3 One proposed solution to the environmental problems caused by transportation vehicles is to use fuel cells powered by hydrogen fuel. In place of the gasoline tank, a pressurized hydrogen tank would be needed. Tank pressures as high as 800 atm have been suggested. Use the Guggenheim equation of state (Guggenheim, E.A., Molecular Physics, 9, pp. 199-200, 1965) to do your calculations and compare the results it provides with a reliable source. The Guggenheim equation of state is given by:

41 where

41a bz y

RT v vy

a.) Determine the constants, a and b, for hydrogen by forcing the pressure-volume isotherm to have a slope of zero and an inflection point at the critical point. Critical property data for hydrogen are provided in EES.

b.) Determine the hydrogen storage volume required to provide a vehicle range equivalent to 15 gallons of gasoline for storage pressures between 100 to 800 atm. Note that the density of gasoline on a mass basis is 0.7 of the density of water at 300 K and 1 atm. The energy content of one kg of gasoline is about 44.4 MJ/kg whereas the energy content of hydrogen is 119.95 MJ/kg.

c.) Compare the critical compressibility predicted by the Guggenheim equation with the value determined from EES property functions or other reported critical point data.


10.A-4 The two-parameter MMM equation of state proposed by Mohsen-Nia et al. (Mohsen-Nia, M., Moddaress, H., and Mansoori, G.A., A Simple Cubic Equation of State for Hydrocarbons and Other Compounds, SPE Paper #26667, Proceedings of the 1993 Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Houston, TX.) is claimed to be more accurate than the Peng-Robinson and Redlich-Kwong-Soave equations. The MMM equation can be expressed in the following form.

3/21.319v b azv b RT v b

where z is the compressibility factor, T_r is the reduced temperature, Tc and Pc are the critical temperature and pressure, respectively, and parameters a and b are determined from by adjusting the parameters determined from critical point information as follows:

2 2.5

3 31 1

3 31 1

1 1

0.486989 0.064662

1 / 1 /1 1

0.036139 0.14167 0.0634 0.18769

c cc c

c c

r rc c

R T RTa bP P

T Ta a b b

where is the acentric factor. a.) Test the MMM equation's ability to accurately predict the specific volume of carbon

dioxide in the superheated, subcooled and saturated regimes by plotting the 240 K, 280 K, and 320 K isotherms over a range of pressures between atmospheric and 275 bar. Compare your results with a reliable source and prepare a short summary of your results indicating the accuracy of the equation in the different regimes.

b.) Apply the requirements that the equation of state must have zero first and second derivatives of P with respect to v at the critical point and use this information to determine parameters ac and bc. Compare your results with the values provided above.


10.A-5 The improved Patal-Teja equation of state (Patel, N.C., Improvements of the Patel-Teja Equation of State, Int. Journal of Thermophysics, Vol. 17, No. 3, 1996) reportedly is more accurate than the Redlich-Kwong-Soave and Peng-Robinson equations of state for liquid density. The equation has the following form.

( )RT a TP

v b v v b c v b

where 2 2

( )c c ca b cc c c

R T RT RTa T b cP P P

1 2 3( ) 1 1 1 1Nr r rT c T c T c T The values of the three dimensionless constants, a , b , and c , are established by required that, at the critical point,

2

20 0c c

ccrit ccrit

P vP P zv v RT

Applying these criteria, the author notes that:

2 21 3

3 3 1 2 1 3c c

a c c b b c

zz z z

b is obtained by solving for the smallest positive root of the following cubic equation 3 2 2 32 3 3 0b c b c b cz z z The purpose of this problem is to test the Patel-Tejas equation's ability to accurately predict PVT behavior in the superheated, subcooled and saturated regimes for carbon dioxide. Patel provides the following constants for carbon dioxide.

c1 = 0.63199 c2 = 2.69935 c3 = 0 N=1

Plot pressure versus specific volume for isotherms at 340 K, 300 K, and 260 K. Overlay the isotherms on a P-v property chart and comment on the agreement with the values provided by EES.


10.A-6 There have been literally hundreds of proposed equations of state for pure fluids. Recently, Lin et al (Lin, H., Duan, Y., Zhang, T, and Huang, Z., Volumetric Property Improvement for the Soave-Redlich-Kwong Equation of State, Ind. Eng. Chem Res, Vol. 45, pp. 1829-1839, 2006) have proposed an equation of state which can be summarized as follows.

( )

-RT a TP

v c b v c v b c

where

2 2

1/ 2 1/ 2 2

( ) 0.42748

1 1- 0.480 1.574 - 0.176

10.08664 ( )3

1 exp 1 for 1( )

1 exp 0.5 for 1

13.7303exp 60.2833 0.23343

3

c

c

r

c cc

c c

r r

r

c

R Ta TP

m T m

RT RTb c z f TP P

T Tf T

T

z

1.4620exp 16.0813 4.09573 c

z

In the above equations, Tr is the reduced temperature, is the acentric factor and zc is the critical compressibility, i.e., the compressibility at the critical point.

a.) Using the Lin equation of state, calculate the specific volume of a hydrocarbon fluid in the superheated, subcooled and saturated regimes. Compare the specific volumes obtained from the equation data from EES for this fluid. In the superheat region, you should select temperatures and pressures that cause the compressibility to significantly differ from unity. Summarize your results in a table and indicate the percentage error.

b.) Plot P vs v along at the critical temperature in the vicinity of the critical point. Use the plot to determine the critical volume for the fluid and compare the result with an accepted value.


10.A-7 A heating plant for a building complex uses natural gas (methane) as a fuel. A concern has been raised regarding an interruption in service. The plant manager has suggested that they stockpile cylinders of methane to be used in an emergency. Each cylinder has a volume of 2 ft3 and, when fully filled, the pressure is 3000 psia at a temperature of 70F. When combusted in the plant equipment, methane provides 18,060 Btu/lbm.

a.) Estimate the number of cylinders of methane needed to supply a heating load of 500,000 Btu/hr for a 24 hour period using the Peng-Robinson equation of state.

b.) Compare the result in part a) with the value obtained using EES property data.


10.A-8 The Carnahan-Starling-DeSantis (CSD) equation of state (Morrison, G and McLinden, M.O., NBS Technical Note 1226, August, 1986) has the following form.

2 3

31 where

41y y y a bZ y

RT v b vy

where Z is the compressibility factor. The claimed advantages of this equation are good predictions of liquid density without the complex parameter fitting procedures of more elaborate equations and the ability to be used in refrigerant mixture calculations. When used for a pure fluid without additional data, the values of a and b must be determined by requiring the critical isotherm to have a slope of zero and an inflection point at the critical point.

Like most equations of state, the CSD equation is explicit in pressure but implicit in specific volume. A numerical method (e.g., Newton's method) is needed to solve for specific volume at specified temperature and pressure. For some conditions, there may be three real solutions for the specific volume. In this case, the smallest and largest solutions are the estimated specific volumes of saturated liquid and vapor, respectively. The intermediate solution corresponds to an unstable state which is physically unrealizable.

In this problem, we will test the CSD equation's ability to accurately predict PVT behavior for R1234yf, which is a new refrigerant with properties similar to R134a. Critical property data for R1234yf can be obtained with the P_crit, T_crit and v_crit property routines in EES. EES also provides preliminary property data for this fluid so that you can compare the results of the CSD equation of state with EES.

a.) Using the critical temperature, pressure, and volume provided by EES, calculate the values of a and b (on a molar basis) for the CSD equation of state and determine the critical compressibility.

b.) Use the CSD equation of state to determine the critical volume at the critical pressure and temperature provided by EES. Note that the values of a and b are needed to calculate the critical volume. These values should be obtained by repeating part a, but use the critical volume obtained from the CSD equation of state in place of the critical volume supplied by EES. Compare the values of a and b and the critical compressibility with the values obtained in part a.

c.) Test the CSD equation of state by calculating the specific volume as a function of pressure for pressures ranging from 1 bar to 50 bar at temperatures of 500 K, 400 K, and 300 K. Compare the results with EES. (Note that at conditions below the critical temperature, you need to select the correct root from the CSD equation. One way to do this is to use the EES specific volume value as a guess for the CSD specific volume.


10.A-9 Properties are commonly formulated in terms of a reduced Helmholtz free energy function, , which can be represented as

( ), u T sT vRT

The reduced Helmholtz equation is a complete equation of state so that all thermodynamic properties can be determined from an equation in this form. The reduced Helmholtz function is ordinarily represented as the sum of an ideal gas and residual component (IG and res, respectively): , , ,IG resT v T v T v This problem illustrates that a pressure explicit equation of state can be converted into the reduced Helmholtz form.

a.) Show that

1, vresTv

RTT v P dvRT v

(1) b.) Using the Peng-Robinson equation of state to show that

1 2

, ln ln2 2 1 2

res

v bv aT vv b RT b v b

(2)


10.B-1 The isentropic index, k, is defined as the coefficient relating pressure and volume such that during an isentropic process, Pvk=Constant. For an ideal gas, k is the ratio of the specific heat capacities, cp/cv. Derive a relation for k that is applicable for non-ideal gas behavior. Your relation should involve only the specific heat ratio and expressions involving pressure, volume, and temperature.


10.B-2 The specific heat of a gas is 825 J/kg-K at 800 K. The molar mass of the gas is 30 kg/kmol. The gas is known to obey the Berthelot equation of state at 800 K for pressures ranging from atmospheric to 100 MPa. The Berthelot equation is

2RT aPv b T v

where a=21420 N-m4-K/kg2 and b=0.00126 m3/kg. Prepare a plot of the constant volume and constant pressure specific heat capacities as a function of pressure for pressures between atmospheric and 100 MPa.


10.B-3 The change in internal energy of ideal gas is identically zero if it is compressed isothermally. Determine the values of U and S if one kmol of ammonia vapor is isothermally compressed from 22.4 liters to 10 liters at 0C, assuming that the gas behavior can be described by the Berthelot equation of state.

2aP v b RTT v

a = 1.725E8 [K-Pa-m6/kmol2] b = 0.03737 m3/kmol


10.B-4 Consider a piece of rubber as a thermodynamic system. The differential work on the rubber is given by

W F dl where F is the force exerted on the rubber when it is extended to a length of l. a.) Assuming a reversible process, show that the fundamental property relation for this

system is dU TdS Fdl . b.) Develop an expression for the differential change in entropy of the system dS in terms

of independent variables T and l. Your result should involve only T, F, l, and Cl,

where Cl, is the heat capacity at constant length, defined as: ll

UCT

.

c.) Derive an equation that will determine the temperature (T) as a function of length (l) for reversible adiabatic stretching process, given the following equation of state for the rubber.

oF bT l l where

b is a positive-valued constant ol is the unstretched length at temperature To. Assume Cl to be constant. d.) Use your relation in c) to determine whether the temperature of the rubber increases

or decreased when it is adiabatically stretched.


10.B-5 Find an expression for the following derivatives involving only P, T, v, cP and cv a.)

u

Tv

b.)

s

hv

c.) u

Ph


10.B-6 Propane at 50 atm is heated at constant pressure from 500 K to 600 K. Determine the heat transfer and entropy change for this process per mole of propane. Assume that the constant pressure specific heat of propane is opc = 125 J/gmol-K and that propane obeys the following equation of state.

2

RT aPv b v

where

a = 9.255 liters2 atm/gmol2 b = 0.09033 liters/gmol


10.B-7 The purpose of this problem is to determine the specific volume, enthalpy, and entropy of carbon dioxide using the Beattie-Bridgeman equation of state equation of state equation of state described in problem 10.A-1. The ideal gas specific heat capacity in units of kJ/kmol-K is

0.5 23.7357 30.529 4.1034 0.024198opc where [ ] /100T K . Write a program to calculate and plot the compressibility, specific enthalpy, and specific entropy of carbon dioxide as a function of pressure for 100 Pa < P < 10e7 Pa for isotherms of 310 K and 350 K. Refer your values of h and s to reference values of 9.211 [kJ/kg] and 0.03123 [kJ/kg-K], respectively at 310 K and 101.3 kPa, which will result in your values having the same reference states as used in EES. Compare your results with values from EES. At what conditions do significant errors occur and what is the major cause of these errors?


10.B-8 A closed system contains two pounds of a gas mixture of unknown composition that is compressed isothermally from 240 to 320 psia at 300F. Experimental P, v, T data are provided in Table 10.B-8. Using these data, estimate the heat, the work and the change in entropy of the gas, assuming the process to be reversible. If these data are insufficient to do the calculation, indicate what other data are required and how you would use the additional data to do the calculation.

Table 10.B-8: Values of specifc volume [ft3/lbm] P [psia] 285F 290F 295F 300F 305F 310F 315F

230 0.491 0.497 0.503 0.509 0.515 0.521 0.527 240 0.465 0.471 0.477 0.483 0.489 0.495 0.500 250 0.441 0.447 0.453 0.459 0.465 0.470 0.476 260 0.419 0.425 0.431 0.437 0.442 0.448 0.453 270 0.398 0.404 0.410 0.416 0.421 0.427 0.432 280 0.379 0.385 0.391 0.396 0.402 0.407 0.413 290 0.361 0.367 0.373 0.378 0.384 0.389 0.395 300 0.344 0.350 0.356 0.361 0.367 0.372 0.377 310 0.328 0.334 0.340 0.345 0.351 0.356 0.361 320 0.313 0.319 0.325 0.330 0.336 0.341 0.346 330 0.299 0.305 0.310 0.316 0.321 0.327 0.332


10.B-9 Write a program to determine the specific volume, enthalpy, and entropy of isobutane using the Lin et al (2006) equation of state described in problem 10.A-6. Critical constants for isobutane are available in EES. The ideal gas specific heat of isobutane can be approximated as

0 2 3 46 772 0 34147 1 0271 4 3 6849 8 2 0429 11 . . . . . pc T E T E T E T where 0pc is in units of kJ/kmol-K and T is in Kelvin. Refer your values of h and s to

reference values of 598.9 [kJ/kg] and 2.513 [kJ/kg-K], respectively at 298.15 K and 101.3 kPa. Use your program to plot the specific volume, enthalpy and entropy of isobutane as a function of pressure at 420 K, 450 K and 500 K for pressures ranging between 0.10 and 10 MPa. Compare your results with values from EES and comment on the differences.


10.B-10 A gas storage tank with a volume of 3 m3 contains a gas at 900 kPa and 25 C, the temperature of the surroundings. Compressibility data for this gas at 25C, 50C and 75C are provided in Figure 10.C-10. A valve on this tank leaks slightly so that, after a considerable time, the tank pressure drops to 300 kPa. a.) Estimate the heat transfer between the tank contents and the surroundings for this

process. b.) What would the heat transfer be if the gas obeyed the ideal gas law? c.) Estimate the change in specific entropy of the gas that remains in the tank.

0 200 400 600 800 10000.7

0.75

0.8

0.85

0.9

0.95

1

P [kPa]

Z

0C

25C

50C

Figure 10.C-10: Compressibility factor versus pressure at 0, 25 and 50C


10.B-11 A superheated organic vapor flowing through an insulated pipeline passes through a restriction. Upstream of the restriction, the pressure is 35 bar and the temperature is 230C. Downstream of the restriction, the pressure is 31.5 bar and the temperature is 225.5C. Pressure, volume, temperature data for this substance are provided in Table 10.B-11.

Table 10.B-11: property data for a superheated organic vapor Pressure

(bars) Specific volume

(m3/kg) 220C 230C 240C

30 0.02302 0.02431 0.0255 35 0.01839 0.01971 0.02089 40 0.01466 0.01611 0.01733

a.) Estimate an average value for the constant pressure specific heat capacity at the conditions encountered at the pipeline restriction.

b.) Estimate the change in specific entropy of the vapor as it passes through the restriction.

c.) It has been reported that in some sections of the pipeline, the temperature increases as it passes through a restriction. Is this possible? Would the entropy change be positive in this case? Explain.


10.B-12 A thermodynamic system consists of a solid paramagnetic substance having a magnetic moment per unit volume, M ([=] amp/m). The substance is subjected to a magnetic field of intensity F ([=] amp/m) and is reversibly magnetized. The work done on the substance is:

oW V F dM where o is the magnetic permeability of free space (4x107 N/amp2) V is the volume of the material in m3. The equilibrium relationship for a paramagnetic solid can be expressed by the Curie

equation as follows:

V M TFmC

where m is the mass of the solid [kg] T is the absolute temperature [K] C is the Curie constant, a characteristic of the material with units of m3-K/kg a.) Assuming that the only work interaction is that due to the magnetization of the

substance, show that:

odU TdS V F dM b.) Develop an expression for dS in terms of independent variables T and M. (Your

result should involve only T, V, o , M, m, C, and Cm, the specific heat of the substance at constant magnetic moment.

c.) Determine an expression for T as a function of M for a reversible adiabatic process. Does the temperature increase or decrease when the substance is demagnetized?


10.B-13 When a pressure disturbance occurs in a compressible fluid, the disturbance travels with a velocity that depends on the state of the fluid. A sound wave is a very small pressure disturbance which can be approximated as an isentropic process. The speed of sound, c, is an easily measured thermodynamic quantity defined by:

s

Pc

where P is pressure, is the fluid density, and s is specific entropy. Derive an equation for the speed of sound through a fluid that is described by the MMM equation of state proposed by Mohsen-Nia et al. which is described in Problem 10.A-4. Your equation should involve only of P, v, T, cp/cv and derivatives involving these properties. Use your equation to calculate and plot the speed of sound through carbon dioxide at 350, 300, and 250 K as a function of reduced pressures between 0.1 and 1.0. Use the EES value of cp/cv in your evaluations. Compare your results with the SoundSpeed function results provided by EES. Provide an explanation for differences between your results and the accepted values.


10.B-14 A large sphere with a pinhole provide a good approximation of a black (i.e., perfect absorber) surface. Energy exchange between the inner surface of the sphere occurs by thermal radiation. Experiments show that if the volume of the sphere (V) is increased with the temperature (T) held constant, the rate of radiation exchange is increased but the energy density e(T) (i.e., the energy per unit volume) remains constant. Thus, the internal energy of the surface of the sphere can be expressed as U = V e(T). Experiments also show that radiation exerts a pressure P=1/3 e(T) on the walls of the surface. Using these experimental facts, derive an equation for e(T) in terms of the temperature T, a reference temperature, To and a constant eo=e(To).


10.B-15 A gas obeys the equation of state

2

RT aPv b T v

where a=1.426e4 kJ-m3 K1/2/kmol2 and b =0.0211 m3/kmol. The ideal gas constant volume specific heat capacity of this gas is 30 kJ/kmol-K. The specific volume of the gas is 0.423 m3/kmol at 400 K. The gas is heated at constant volume to 800 K. a.) Determine the initial and final pressures b.) Determine the change in internal energy per kmol of gas.


10.B-16 The specific heat capacity ratio, k, is needed to calculate the speed of sound and for other thermodynamic processes. This problem investigates the use of CSD equation of state described in problem 10.A-8 for calculating the specific heat ratio of refrigerant R1234yf.

a.) Derive a general relation for vT

cv

b.) In the temperature range between 300 K and 500 K, the constant volume specific

heat of R1234yf at ideal gas conditions in J/kmol-K can be represented by: 26816.44 368.528 0.234306ovc T T

Specialize the general relation found in part a.) to allow evaluation of cv for R1234yf at a specified temperature and pressure using the CSD equation of state.

c.) Use the relation for cv determined in part b and the general relation between cp and cv to determine the specific heat ratio. Calculate and plot the specific heat ratio as a function of pressure for pressures between 1 bar and 50 bar at 400 K and 500 K. Compare the result with EES and comment on the agreement.


10.B-17 High accuracy equations of state for many substances have recently been formulated by representing the reduced Helmholtz Free energy function as the sum of ideal gas and residual parts, i.e.,

, , ,IG resaRT (1)

where crit

(2)

critTT

(3) a is the specific Helmholtz free energy and superscripts IG and res refer to the ideal gas and residual components. For R134a, the ideal gas and residual components have been correlated as:

51 2 34

, ln ln inIG ii

a a a a

(4) 8 21

1 9, expi i i i id t d t eres i i

i ia a

(5)

Coefficients needed in Eqs. (4) and (5) for R134a are provided in the table 10.B-17 and in EES Lookfile FEQR134a.lkt.

a.) Derive expressions for the pressure, specific internal energy, and specific entropy as functions of and .

b) Using your results from part a.), calculate values of pressure, specific enthalpy, and specific entropy at 300 K as a function of specific volume ranging from 1 m3/kg to 0.03 m3/kg. Compare your results with values from EES. Note that EES uses the ASHRAE reference for h and s whereas the Tillner-Roth equation of state uses the IIR reference state. Include the following directive in your program to change the reference state in EES.

Table 10.B-17: Coefficients for Eqs. (4) and (5) for R134a from Tillner-Roth, R., Fundamental Equations of State, Shaker Verlag, Aachen, 1998.

oia in ia ie it id

1 -1.019535 0 0.49822300 0 0 1 2 9.047135 0 0.02458698 0 0 3 3 -1.629789 0 0.00085701 0 0 6 4 -9.723916 -0.5 0.00047886 0 1.5 6 5 -3.927170 -0.75 -1.80080800 0 1.5 1 6 0 0 0.26716410 0 2 1 7 -0.04781652 0 2 2 8 0.05586817 0 -0.5 2 9 0.33240620 1 3 2 10 -0.00748591 1 5 2 11 0.01423987 1 1 5 12 0.20571440 2 6 1


13 -0.51845670 2 5 1 14 -0.08692288 2 5 4 15 0.00010173 2 1 4 16 0.00046033 2 10 4 17 -0.00500046 2 10 2 18 -0.00349784 3 10 1 19 0.00699504 3 18 5 20 -0.01452184 3 22 3 21 -0.00012855 4 50 10


10.B-18 Calculate plot the change in enthalpy and entropy as a function of initial pressure using the Peng-Robinson equation of state for carbon dioxide undergoing an isothermal change in pressure from 10,000 to 100 kPa at 310 K. You may use the Peng-Robinson library functions provided in EES. Compare your result with accepted values.


10.C-1 A 180 lbm person is planing to ice skate on hollow ground blades that have a total area in contact with the ice of 0.012 in2. The ice temperature is 28F. Will the ice melt under the blades? Data for water at its triple point are provided in Table 10.C-1.

Table 10.C-1 Triple point data for water Pressure 611.7 Pa Temperature 273.16 K liquid specific volume 0.001000 m3/kg solid specific volume 0.001091 m3/kg liquid specific enthalpy 23.26 J/kg solid specific enthlapy -333,316 J/kg


10.C-2 The critical temperature and pressure of n-butane are 425.2 K and 3796 kPa, respectively. Its molar mass is 58.12 kg/kmol. In the temperature range between 300 K and 400 K, the vapor pressure of n-butane is represented by

ln( ) 21.54 2722.08 /satP T where satP is in Pa and T is in K. The specific volume of liquid n-butane in this temperature range is approximately

20.00535 0.00002577 4.608E-8liqv T T where liqv is in units of m

3/kg. Using these data, estimate the enthalpy of vaporization of n-butane liquid at 340 K with a.) The Clausius-Clapeyron equation b.) The Clapeyron equation c.) Compare the results to each other and to the value obtained from EES.


10.C-3 Shown in Figure 10.C-3 is a plot of pressure versus specific volume along an isotherm of 90C for R134a calculated with the Peng-Robinson equation of state. Using only this plot, estimate the vapor pressure of R134a at 90C. Compare your result with the vapor pressure provided by EES.

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.00825

30

35

40

v [m3/kg]

P [b

ar]

Peng-Robinson Equation of State for R134a

T=90C

Figure 10.C-3 Pressure versus volume at 90C for R134a determined with the Peng Robinson EOS


10.C-4 The saturation pressure, liquid density and vapor density of a refrigerant have been measured as a function of temperature. These data are reported in Table 10.C-4.

Table 10.C-4: Measured saturation data for a refrigerant T

(C) P

(bar) liquid density

(kg/m3) Vapor density

(kg/m3) 50 2.346 1414 12.88 55 2.707 1402 14.75 60 3.11 1389 16.82 65 3.557 1375 19.11 70 4.05 1362 21.64 75 4.594 1348 24.42 80 5.191 1334 27.48 85 5.844 1320 30.84 90 6.556 1306 34.52 95 7.332 1291 38.54 100 8.174 1276 42.94

a.) Calculate and plot the specific enthalpy change of vaporization and the specific entropy change of vaporization as a function of temperature.

b.) Estimate the normal boiling point for this refrigerant.


10.C-5 Data for an organic substance that is used in an industrial process are reported in Table 10.C-5.

Table 10.C-5: Physical property data for an organic substance Molecular weight 127.6 kg/kmol Normal melting point 72.5C Heat of fusion at 1 atm 19.674E6 J/kmol Vapor pressure at the normal melting point 5.0 mm Hg Vapor pressure at 100C 20.0 mm Hg Density of solid at the normal melting point 1.429 g/cm3 Density of liquid at the normal melting point 1.150 g/cm3

Using these data and appropriate assumptions, a.) estimate the heat of vaporization of the liquid at 1 atm b.) esimate the heat of sublimation of the solid at 1 atm c.) estimate the melting point of the solid at 100 atm d.) estimate the boiling point of the liquid at 1 atm.


10.C-6 Liquid benzene is pumped to a vaporizer as a saturated liquid at a 2200 kPa. Benzene vapor exits the vaporizer as a wet vapor with 92% quality and essentially the same pressure as it entered. Estimate the heat load on the vaporizer per kg of benzene. Data for benzene are provided in Table 10.C-6.

Table 10.C-6: Property Information for Benzene T

[C] Psat

[kPa] liq

[kg/m3] 220 1958.7 622.2 225 2107.8 612.8 230 2265.2 602.9 235 2431.2 592.5 240 2606.0 581.6 245 2790.2 570.1

Critical Temperature = 562.0 K Critical Pressure = 4.894 Mpa

Acentric factor =0.2092 Molar mass = 78.108 kg/kmol


10.C-7 Refrigerant R134a at 300 K is isothermally compressed from as initial pressure of 100 kPa until it reaches a condition of saturation. Use the Peng Robinson equation of state to determine the following quantities for this process: a.) the saturation pressure at 300 K by equating the fugacities of the liquid and vapor b.) the change in specific volume c.) the change in specific enthalpy d.) the change in specifi entropy

You are welcome to use the Peng-Robinson library in EES. Compare your values with the R134a property data.


10.C-8 The vapor pressure of water can be represented by the equation:

5301ln 32.99 1.2236ln( )P TT

where P is the saturation pressure in Pa and T is in K. a.) Using only the information given above, prepare a plot of the enthalpy of

vaporization of water as a function of temperature for temperatures between 300 K and 635 K. Compare the results with the steam table data.

b.) Use the specific volume data provided in the steam tables to obtain a more correct result.


10.C-9 The vapor pressure of carbon tetrachloride (CCl4, MW=153.82) at several temperatures is shown in Table 10.C-9. Using these data, estimate the enthalpy of vaporization of carbon tetrachloride in this temperature range.

Table 10.C-9: Vapor pressure versus temperature for carbon tetrachloride T [C] 25 35 45 55 P [atm] 0.15 0.229 0.341 0.492


10.C-10 Using the Peng Robinson equation of state, estimate the saturation vapor pressure of carbon dioxide at 250 K and the corresponding specific volumes of saturated liquid and vapor by: a.) using the Fundamental property relation involving Gibbs free energy, and by b.) equating the fugacities of saturated liquid and vapor.

Compare these estimates with values from a respected source.


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