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physics
thermodynamics

Fundamentals of Thermodynamics 6th edition Richard E. Sonntag, Claus Borgnakke, Gordon J. Van Wylen - Solutions

If I have air at 100 kPa and a) –10oC b) 45oC and c) 110oC what is the maximum absolute humidity I can have?
Explain in words what the absolute and relative humidity expresses?
I want to bring air at 35oC, Φ = 40% to a state of 25oC, ω = 0.01 do I need to add or subtract water?
Mention two uses of the Clapeyron equation.
A mixture of 60% ethylene and 40% acetylene by moles is at 6 MPa, 300 K. The mixture flows through a preheater where it is heated to 400 K at constant P. Using the Redlich Kwong equation of state with a, b for a mixture find the inlet specific volume. Repeat using Kays rule and the generalized
For the previous problem, find the specific heat transfer using Kay’s rule and the generalized charts.
One kmol/s of saturated liquid methane, CH4, at 1 MPa and 2 kmol/s of ethane, C2H6, at 250°C, 1 MPa are fed to a mixing chamber with the resultant mixture exiting at 50°C, 1 MPa. Assume that Kay’s rule applies to the mixture and determine the heat transfer in the process.
A piston/cylinder initially contains propane at T = -7°C, quality 50%, and volume 10L. A valve connecting the cylinder to a line flowing nitrogen gas at T = 20°C, P = 1 MPa is opened and nitrogen flows in. When the valve is closed, the cylinder contains a gas mixture of 50% nitrogen, 50% propane
Consider the following reference state conditions: the entropy of real saturated liquid methane at −100°C is to be taken as 100 kJ/kmol K, and the entropy of hypothetical ideal gas ethane at −100°C is to be taken as 200 kJ/kmol K. Calculate the entropy per kmol of a real gas mixture of 50%
A cylinder/piston contains a gas mixture, 50% CO2 and 50% C2H6 (mole basis) at 700 kPa, 35°C, at which point the cylinder volume is 5 L. The mixture is now compressed to 5.5 MPa in a reversible isothermal process. Calculate the heat transfer and work for the process, using the following model for
A cylinder/piston contains a gas mixture, 50% CO2 and 50% C2H6 (mole basis) at 700 kPa, 35°C, at which point the cylinder volume is 5 L. The mixture is now compressed to 5.5 MPa in a reversible isothermal process. Calculate the heat transfer and work for the process, using the following model for
Consider a straight line connecting the point P = 0, Z = 1 to the critical point P = PC, Z = ZC on a Z versus P compressibility diagram. This straight line will be tangent to one particular isotherm at low pressure. The experimentally determined value is about 0.8 TC. Determine what value of
A 200-L rigid tank contains propane at 400 K, 3.5 MPa. A valve is opened, and propane flows out until half the initial mass has escaped, at which point the valve is closed. During this process the mass remaining inside the tank expands according to the relation Pv1.4 = constant. Calculate the heat
A newly developed compound is being considered for use as the working fluid in a small Rankine-cycle power plant driven by a supply of waste heat. Assume the cycle is ideal, with saturated vapor at 200°C entering the turbine and saturated liquid at 20°C exiting the condenser. The only properties
If I raise the pressure in a solid at constant T, does s go up or down?
A cylinder fitted with a movable piston contains propane, initially at 67oC and 50 % quality, at which point the volume is 2 L. The piston has a cross-sectional area of 0.2 m2. The external force on the piston is now gradually reduced to a final value of 85 kN, during which process the propane
One kilogram per second water enters a solar collector at 40°C and exits at 190°C, as shown in Fig. P13.111. The hot water is sprayed into a direct-contact heat exchanger (no mixing of the two fluids) used to boil the liquid butane. Pure saturated-vapor butane exits at the top at 80°C and is fed
A piston/cylinder contains ethane gas, initially at 500 kPa, 100 L, and at ambient temperature, 0°C. The piston is now moved, compressing the ethane until it is at 20°C, with a quality of 50%. The work required is 25% more than would have been required for a reversible polytrophic process between
An experiment is conducted at −100°C inside a rigid sealed tank containing liquid R-22 with a small amount of vapor at the top. When the experiment is done the container and the R-22 warms up to room temperature of 20°C. What is the pressure inside the tank during the experiment? If the
The refrigerant R-152a, difluoroethane, is tested by the following procedure. A 10-L evacuated tank is connected to a line flowing saturated-vapor R-152a at 40°C. The valve is then opened, and the fluid flows in rapidly, so that the process is essentially adiabatic. The valve is to be closed when
Carbon dioxide gas enters a turbine at 5 MPa, 100°C, and exits at 1 MPa. If the isentropic efficiency of the turbine is 75%, determine the exit temperature and the second-law efficiency.
A 4- m3 un insulated storage tank, initially evacuated, is connected to a line flowing ethane gas at 10 MPa, 100°C. The valve is opened, and ethane flows into the tank for a period of time, after which the valve is closed. Eventually, the whole system cools to ambient temperature, 0°C, at which
A 10- m3 storage tank contains methane at low temperature. The pressure inside is 700 kPa, and the tank contains 25% liquid and 75% vapor, on a volume basis. The tank warms very slowly because heat is transferred from the ambient. a. What is the temperature of the methane when the pressure reaches
A gas mixture of a known composition is frequently required for different purposes, e.g., in the calibration of gas analyzers. It is desired to prepare a gas mixture of 80% ethylene and 20% carbon dioxide (mole basis) at 10 MPa, 25°C in an un insulated, rigid 50-L tank. The tank is initially to
Determine the heat transfer and the net entropy change in the previous problem. Use the initial pressure of the carbon dioxide to be 4.56 MPa before the ethylene is flowing into the tank. A gas mixture of a known composition is frequently required for different purposes, e.g., in the calibration of
As P → 0, the specific volume v → ∞. For P → ∞, does v → 0?
Must an equation of state satisfy the two conditions in Eqs. 13.50 and 13.51?
What is the benefit of the generalized charts? Which properties must be known besides the charts themselves?
What does it imply if the compressibility factor is larger than 1?
The slope dP/dT of the vaporization line is finite as you approach the critical point, yet hfg and vfg both approach zero. How can that be?
A special application requires R-12 at −140°C. It is known that the triple-point temperature is −157°C. Find the pressure and specific volume of the saturated vapor at the required condition.
Ice (solid water) at −3°C, 100 kPa, is compressed isothermally until it becomes liquid. Find the required pressure.
An approximation for the saturation pressure can be In Psat = A – B/T, where A and B are constants. Which phase transition is that suitable for, and what kind of property variations are assumed?
In a Carnot heat engine, the heat addition changes the working fluid from saturated liquid to saturated vapor at T, P. The heat rejection process occurs at lower temperature and pressure (T − ΔT), (P − ΔP). The cycle takes place in a piston cylinder arrangement where the work is boundary
Calculate the values hfg and sfg for nitrogen at 70 K and at 110 K from the Clapeyron equation, using the necessary pressure and specific volume values from Table B.6.1.
Ammonia at –70oC is used in a special application at a quality of 50%. Assume the only table available is B.2 that goes down to –50oC. To size a tank to hold 0.5 kg with x = 0.5, give your best estimate for the saturated pressure and the tank volume.
The saturation pressure can be approximated as In Psat = A – B/T, where A and B are constants. Use the steam tables and determine A and B from properties at 25oC only. Use the equation to predict the saturation pressure at 30oC and compare to table value.
Using the properties of water at the triple point, develop an equation for the saturation pressure along the fusion line as a function of temperature.
Helium boils at 4.22 K at atmospheric pressure, 101.3 kPa, with hfg = 83.3 kJ/kmol. By pumping a vacuum over liquid helium, the pressure can be lowered, and it may then boil at a lower temperature. Estimate the necessary pressure to produce a boiling temperature of 1 K and one of 0.5 K.
In view of Clapeyron’s equation and Fig. 3.7, is there something special about ice I versus the other forms of ice?
A certain refrigerant vapor enters a steady flow constant pressure condenser at 150 kPa, 70°C, at a rate of 1.5 kg/s, and it exits as saturated liquid. Calculate the rate of heat transfer from the condenser. It may be assumed that the vapor is an ideal gas, and also that at saturation, vf
Using thermodynamic data for water from Tables B.1.1 and B.1.5, estimate the freezing temperature of liquid water at a pressure of 30 MPa.
Small solid particles formed in combustion should be investigated. We would like to know the sublimation pressure as a function of temperature. The only information available is T, hFG for boiling at 101.3 kPa and T, hIF for melting at 101.3 kPa. Develop a procedure that will allow a determination
A container has a double wall where the wall cavity is filled with carbon dioxide at room temperature and pressure. When the container is filled with a cryogenic liquid at 100 K the carbon dioxide will freeze so that the wall cavity has a mixture of solid and vapor carbon dioxide at the sublimation
Use Gibbs relation du = Tds – Pdv and one of Maxwell’s relations to find an expression for (∂u/∂P)T that only has properties P, v and T involved. What is the value of that partial derivative if you have an ideal gas?
Start from Gibbs relation dh = Tds + vdP and use one of Maxwell’s equation to get (∂h/∂v)T in terms of properties P, v and T. Then use Eq.13.24 to also find an expression for (∂h/∂T)v.
From Eqs. 13.23 and 13.24 and the knowledge that Cp > Cv what can you conclude about the slopes of constant v and constant P curves in a T-s diagram? Notice that we are looking at functions T(s, P or v given).
Derive expressions for (∂T/∂v)u and for (∂h/∂s)v that do not contain the properties h, u, or s. Use Eq. 13.30 with du = 0.
Develop an expression for the variation in temperature with pressure in a constant entropy process, (∂T/∂P)s, that only includes the properties P–v–T and the specific heat, Cp. Follow the development for Eq.13.32.
Use Eq. 13.34 to get an expression for the derivative (∂T/∂v)s. What is the general shape of a constant s process curve in a T-v diagram? For an ideal gas can you say a little more about the shape?
Evaluate the isothermal changes in the internal energy, the enthalpy and the entropy for an ideal gas. Confirm the results in Chapters 5 and 8. We need to evaluate duT, dhT and dsT for an ideal gas: P = RT/v.
Determine the volume expansivity, αP, and the isothermal compressibility, βT, for water at 20°C, 5 MPa and at 300°C, and 15 MPa using the steam tables.
What are the volume expansivity αp, the isothermal compressibility βT, and the adiabatic compressibility βs for an ideal gas?
Find the speed of sound for air at 20°C, 100 kPa using the definition in Eq. 13.43 and relations for polytrophic processes in ideal gases.
Assume a substance has uniform properties in all directions with V = LxLyLz and show that volume expansivity αp = 3δT. Hint: differentiate with respect to T and divide by V. V = LxLyLz
A cylinder fitted with a piston contains liquid methanol at 20°C, 100 kPa and volume 10 L. The piston is moved, compressing the methanol to 20 MPa at constant temperature. Calculate the work required for this process. The isothermal compressibility of liquid methanol at 20°C is 1.22 × 10-9 m2/N.
Use Eq. 13.32 to solve for (∂T/∂P)s in terms of T, v, Cp and αp. How large a temperature change does 25oC water (αp = 2.1 × 10-4 K-1) have, when compressed from 100 kPa to 1000 kPa in an isentropic process?
Sound waves propagate through a media as pressure waves that cause the media to go through isentropic compression and expansion processes. The speed of sound c is defined by c2 = (∂P/∂ρ)s and it can be related to the adiabatic compressibility, which for liquid ethanol at 20°C is 9.4 × 10-10
For commercial copper at 25oC (see table A.3) the speed of sound is about 4800 m/s. What is the adiabatic compressibility βs?
Consider the speed of sound as defined in Eq. 13.43. Calculate the speed of sound for liquid water at 20°C, 2.5 MPa, and for water vapor at 200°C, 300 kPa, using the steam tables.
Sketch on a P-T diagram how a constant v line behaves in the compressed liquid region, the two-phase L-V region and the superheated vapor region?
Soft rubber is used as a part of a motor mounting. Its adiabatic bulk modulus is Bs = 2.82 × 106 kPa, and the volume expansivity is αp = 4.86 × 10-4 K-1. What is the speed of sound vibrations through the rubber, and what is the relative volume change for a pressure change of 1 MPa?
Liquid methanol at 25oC has an adiabatic compressibility of 1.05 × 10-9 m2/N. What is the speed of sound? If it is compressed from 100 kPa to 10 MPa in an insulated piston/cylinder, what is the specific work?
Use Eq. 13.32 to solve for (∂T/∂P)s in terms of T, v, Cp and αp. How much higher does the temperature become for the compression of the methanol in Problem 13.51? Use αp = 2.4 × 10-4 K-1 for methanol at 25oC.
Use the equation of state as shown in Example 13.3 where changes in enthalpy and entropy were found. Find the isothermal change in internal energy in a similar fashion; do not compute it from enthalpy.
Evaluate changes in an isothermal process for u, h and s for a gas with an equation of state as P (v − b) = RT.
Two un insulated tanks of equal volume are connected by a valve. One tank contains a gas at a moderate pressure P1, and the other tank is evacuated. The valve is opened and remains open for a long time. Is the final pressure P2 greater than, equal to, or less than P1/2? Hint: Recall Fig. 13.5.
Determine the reduced Boyle temperature as predicted by an equation of state (the experimentally observed value for most substances is about 2.5), using the van der Waals equation and the Redlich–Kwong equation. Note: It is helpful to use Eqs. 13.47 and 13.48 in addition to Eq. 13.46
Develop expressions for isothermal changes in internal energy, enthalpy and entropy for a gas obeying the van der Waals equation of state.
Develop expressions for isothermal changes in internal energy, enthalpy and entropy for a gas obeying Redlich-Kwong equation of state.
Consider the following equation of state, expressed in terms of reduced pressure and temperature: Z = 1 + (Pr/14Tr)[1 – 6T−2r]. What does this predict for the reduced Boyle temperature?
What is the Boyle temperature for the following equation of state: P = RT/v-b – a/v2T where a and b are constants.
Show that the van der Waals equation can be written as a cubic equation in the compressibility factor involving the reduced pressure and reduced temperature as
Determine the second virial coefficient B(T) using the van der Waals equation of state. Also find its value at the critical temperature where the experimentally observed value is about –0.34 RTc/Pc.
Determine the second virial coefficient B(T) using the Redlich-Kwong equation of state. Also find its value at the critical temperature where the experimentally observed value is about –0.34 RTc/Pc.
One early attempt to improve on the van der Waals equation of state was an expression of the form P = RT/v-b – a/v2T Solve for the constants a, b, and vC using the same procedure as for the van der Waals equation.
Calculate the difference in internal energy of the ideal-gas value and the real-gas value for carbon dioxide at the state 20°C, 1 MPa, as determined using the virial equation of state, including second virial coefficient terms. For carbon dioxide we have: B = -0.128 m3/kmol, T(dB/dT) = 0.266
Calculate the difference in entropy of the ideal-gas value and the real-gas value for carbon dioxide at the state 20°C, 1 MPa, as determined using the virial equation of state. Use numerical values given in Problem 13.65.
A rigid tank contains 1 kg oxygen at 160 K, 4 MPa. Determine the volume of the tank assuming we can use the Redlich-Kwong equation of state for oxygen. Compare the result with the ideal gas law.
A flow of oxygen at 230 K, 5 MPa is throttled to 100 kPa in a steady flow process. Find the exit temperature and the specific entropy generation using Redlich-Kwong equation of state and ideal gas heat capacity. Notice that this becomes iterative due to the nonlinearity coupling h, P, v and T.
A 200-L rigid tank contains propane at 9 MPa, 280°C. The propane is then allowed to cool to 50°C as heat is transferred with the surroundings. Determine the quality at the final state and the mass of liquid in the tank, using the generalized compressibility chart, Fig. D.1.
If I raise the pressure in an isothermal process does h go up or down for a liquid or solid? What do you need to know if it is a gas phase?
A rigid tank contains 5 kg of ethylene at 3 MPa, 30°C. It is cooled until the ethylene reaches the saturated vapor curve. What is the final temperature?
Refrigerant-123, dichlorotrifluoroethane, which is currently under development as a potential replacement for environmentally hazardous refrigerants, undergoes an isothermal steady flow process in which the R-123 enters a heat exchanger as saturated liquid at 40°C and exits at 100 kPa.
An ordinary lighter is nearly full of liquid propane with a small amount of vapor, the volume is 5 cm3, and temperature is 23°C. The propane is now discharged slowly such that heat transfer keeps the propane and valve flow at 23°C. Find the initial pressure and mass of propane and the total heat
A piston/cylinder contains 5 kg of butane gas at 500 K, 5 MPa. The butane expands in a reversible polytropic process to 3 MPa, 460 K. Determine the polytrophic exponent n and the work done during the process.
Calculate the heat transfer during the process described in Problem 13.73.
A cylinder contains ethylene, C2H4, at 1.536 MPa, −13°C. It is now compressed in a reversible isobaric (constant P) process to saturated liquid. Find the specific work and heat transfer.
Carbon dioxide collected from a fermentation process at 5°C, 100 kPa should be brought to 243 K, 4 MPa in a steady flow process. Find the minimum amount of work required and the heat transfer. What devices are needed to accomplish this change of state?
Consider the following equation of state, expressed in terms of reduced pressure and temperature: Z = 1 + Pr/14 Tr (1 – 6/Tr2 ) What does this equation predict for enthalpy departure from the ideal gas value at the state Pr = 0.4, Tr = 0.9 ?
Consider the following equation of state, expressed in terms of reduced pressure and temperature:Z = 1 + Pr/14 Tr (1 – 6/Tr2 )What does this equation predict for entropy departure from the ideal gas value at the state Pr = 0.4, Tr = 0.9 ?
A flow of oxygen at 230 K, 5 MPa is throttled to 100 kPa in a steady flow process. Find the exit temperature and the entropy generation.
A cylinder contains ethylene, C2H4, at 1.536 MPa, −13°C. It is now compressed isothermally in a reversible process to 5.12 MPa. Find the specific work and heat transfer.
Saturated vapor R-22 at 30°C is throttled to 200 kPa in a steady flow process. Calculate the exit temperature assuming no changes in the kinetic energy, using the generalized charts, Fig. D.2 and the R-22 tables, Table B.4.
250-L tank contains propane at 30°C, 90% quality. The tank is heated to 300°C. Calculate the heat transfer during the process.
The new refrigerant fluid R-123 (see Table A.2) is used in a refrigeration system that operates in the ideal refrigeration cycle, except the compressor is neither reversible nor adiabatic. Saturated vapor at -26.5°C enters the compressor and superheated vapor exits at 65°C. Heat is rejected from
An un insulated piston/cylinder contains propene, C3H6, at ambient temperature, 19°C, with a quality of 50% and a volume of 10 L. The propene now expands very slowly until the pressure in the cylinder drops to 460 kPa. Calculate the mass of propene, the work, and heat transfer for this process.
A geothermal power plant on the Raft River uses isobutene as the working fluid. The fluid enters the reversible adiabatic turbine, as shown in Fig. P13.42, at 160°C, 5.475 MPa, and the condenser exit condition is saturated liquid at 33°C. Isobutene has the properties Tc= 408.14 K, Pc= 3.65 MPa,
A line with a steady supply of octane, C8H18, is at 400°C, 3 MPa. What is your best estimate for the availability in a steady flow setup where changes in potential and kinetic energies may be neglected?
An insulated cylinder fitted with a frictionless piston contains saturated-vapor carbon dioxide at 0oC, at which point the cylinder volume is 20 L. The external force on the piston is now slowly decreased, allowing the carbon dioxide to expand until the temperature reaches - 30oC. Calculate the

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