12–1C Consider the function z(x, y). Plot a differential surface on x-y-z coordinates and indicate dx, dx, dy, dy,( dz)x, (dz)y, and dz. Get 12.1 exercise solution
12–2C What is the difference between partial differentials and ordinary differentials? Get 12.2 exercise solution
12–3C Consider the function z(x, y), its partial derivatives (dz/dx)y and (dz/dy)x, and the total derivative dz/dx. (a) How do the magnitudes (dx)y and dx compare? (b) How do the magnitudes (dz)y and dz compare? (c) Is there any relation among dz,( dz)x, and (dz)y? Get 12.3 exercise solution
12–4C Consider a function z(x, y) and its partial derivative (dz/dy)x. Under what conditions is this partial derivative equal to the total derivative dz/dy? Get 12.4 exercise solution
12–5C Consider a function z(x, y) and its partial derivative (dz/dy)x. If this partial derivative is equal to zero for all values of x, what does it indicate? Get 12.5 exercise solution
12–6C Consider a function z(x, y) and its partial derivative (dz/dy)x. Can this partial derivative still be a function of x? Get 12.6 exercise solution
12–7C Consider a function f(x) and its derivative df/dx. Can this derivative be determined by evaluating dx/df and taking its inverse? Get 12.7 exercise solution
12–8 Consider air at 400 K and 0.90 m3/kg. Using Eq. 12–3, determine the change in pressure corresponding to an increase of (a) 1 percent in temperature at constant specific volume, (b) 1 percent in specific volume at constant temperature, and (c) 1 percent in both the temperature and specific volume. Get 12.8 exercise solution
12–9 Repeat Problem 12–8 for helium. Get 12.9 exercise solution
12–10 Prove for an ideal gas that (a) the P = constant lines on a T-v diagram are straight lines and (b) the high-pressure lines are steeper than the low-pressure lines. Get 12.10 exercise solution
12–11 Derive a relation for the slope of the v = constant lines on a T-P diagram for a gas that obeys the van der Waals equation of state. Get 12.11 exercise solution
12–12 Nitrogen gas at 400 K and 300 kPa behaves as an ideal gas. Estimate the cp and cv of the nitrogen at this state, using enthalpy and internal energy data from Table A–18, and compare them to the values listed in Table A–2b. Get 12.12 exercise solution
12–13E Nitrogen gas at 600 R and 30 psia behaves as an ideal gas. Estimate the cp and cv of the nitrogen at this state, using enthalpy and internal energy data from Table A–18E, and compare them to the values listed in Table A–2Eb. Get 12.13 exercise solution
12–14 Consider an ideal gas at 400 K and 100 kPa. As a result of some disturbance, the conditions of the gas change to 404 K and 96 kPa. Estimate the change in the specific volume of the gas using (a) Eq. 12–3 and (b) the ideal-gas relation at each state. Get 12.14 exercise solution
12–15 Using the equation of state P(v - a) = RT, verify (a) the cyclic relation and (b) the reciprocity relation at constant v. Get 12.15 exercise solution
12–16 Verify the validity of the last Maxwell relation (Eq. 12–19) for refrigerant-134a at 80°C and 1.2 MPa. Get 12.16 exercise solution
12–17 Reconsider Prob. 12–16. Using EES (or other) software, verify the validity of the last Maxwell relation for refrigerant-134a at the specified state. Get 12.17 exercise solution
12–18E Verify the validity of the last Maxwell relation (Eq. 12–19) for steam at 800°F and 400 psia. Get 12.18 exercise solution
12–19 Using the Maxwell relations, determine a relation for (ds/dP)T for a gas whose equation of state is P(v - b) = RT. Get 12.19 exercise solution
12–20 Using the Maxwell relations, determine a relation for (ds/dv)T for a gas whose equation of state is (P - a/v2) (v - b) = RT. Get 12.20 exercise solution
12–21 Using the Maxwell relations and the ideal-gas equation of state, determine a relation for (ds/dv)T for an ideal gas. Get 12.21 exercise solution
12–22C What is the value of the Clapeyron equation in thermodynamics? Get 12.22 exercise solution
12–23C Does the Clapeyron equation involve any approximations, or is it exact? Get 12.23 exercise solution
12–24C What approximations are involved in the ClapeyronClausius equation? Get 12.24 exercise solution
12–25 Using the Clapeyron equation, estimate the enthalpy of vaporization of refrigerant-134a at 40°C, and compare it to the tabulated value. Get 12.25 exercise solution
12–26 Reconsider Prob. 12–25. Using EES (or other) software, plot the enthalpy of vaporization of refrigerant-134a as a function of temperature over the temperature range -20 to 80°C by using the Clapeyron equation and the refrigerant-134a data in EES. Discuss your results. Get 12.26 exercise solution
12–27 Using the Clapeyron equation, estimate the enthalpy of vaporization of steam at 300 kPa, and compare it to the tabulated value. Get 12.27 exercise solution
12–28 Calculate the hfg and sfg of steam at 120°C from the Clapeyron equation, and compare them to the tabulated values. Get 12.28 exercise solution
12–29E Determine the hfg of refrigerant-134a at 50°F on the basis of (a) the Clapeyron equation and (b) the Clapeyron-Clausius equation. Compare your results to the tabulated hfg value. Get 12.29 exercise solution
12–30 Plot the enthalpy of vaporization of steam as a function of temperature over the temperature range 10 to 200°C by using the Clapeyron equation and steam data in EES. Get 12.30 exercise solution
12–31 Using the Clapeyron-Clausius equation and the triplepoint data of water, estimate the sublimation pressure of water at -30°C and compare to the value in Table A–8. Get 12.31 exercise solution
12–32C Can the variation of specific heat cp with pressure at a given temperature be determined from a knowledge of Pv-T data alone? Get 12.32 exercise solution
12–33 Show that the enthalpy of an ideal gas is a function of temperature only and that for an incompressible substance it also depends on pressure. Get 12.33 exercise solution
12–34 Derive expressions for (a) du,( b) dh, and (c) ds for a gas that obeys the van der Waals equation of state for an isothermal process. Get 12.34 exercise solution
12–35 Derive expressions for (a) du,( b) dh, and (c) ds for a gas whose equation of state is P(v - a) = RT for an isothermal process. Get 12.35 exercise solution
12–36 Derive expressions for (du/dP)T and (dh/dv)T in terms of P, v, and T only. Get 12.36 exercise solution
12–37 Derive an expression for the specific-heat difference cp = cv for (a) an ideal gas, (b) a van der Waals gas, and (c) an incompressible substance. Get 12.37 exercise solution
12–38 Estimate the specific-heat difference cp = cv for liquid water at 15 MPa and 80°C. Get 12.38 exercise solution
12–39E Estimate the specific-heat difference cp = cv for liquid water at 1000 psia and 150°F. Get 12.39 exercise solution
12–40 Derive a relation for the volume expansivity b and the isothermal compressibility a (a) for an ideal gas and (b) for a gas whose equation of state is P(v - a) = RT. Get 12.40 exercise solution
12–41 Estimate the volume expansivity b and the isothermal compressibility a of refrigerant-134a at 200 kPa and 30°C. Get 12.41 exercise solution
12–42C What does the Joule-Thomson coefficient represent? Get 12.42 exercise solution
12–43C Describe the inversion line and the maximum inversion temperature. Get 12.43 exercise solution
12–44C The pressure of a fluid always decreases during an adiabatic throttling process. Is this also the case for the temperature? Get 12.44 exercise solution
12–45C Does the Joule-Thomson coefficient of a substance change with temperature at a fixed pressure? Get 12.45 exercise solution
12–46C Will the temperature of helium change if it is throttled adiabatically from 300 K and 600 kPa to 150 kPa? Get 12.46 exercise solution
12–47 Consider a gas whose equation of state is P(v - a) = RT, where a is a positive constant. Is it possible to cool this gas by throttling? Get 12.47 exercise solution
12–48 Derive a relation for the Joule-Thomson coefficient and the inversion temperature for a gas whose equation of state is (P + a/v2)v = RT. Get 12.48 exercise solution
12–49 Estimate the Joule-Thomson coefficient of steam at (a) 3 MPa and 300°C and (b) 6 MPa and 500°C. Get 12.49 exercise solution
12–50E Estimate the Joule-Thomson coefficient of nitrogen at (a) 200 psia and 500 R and (b) 2000 psia and 400 R. Use nitrogen properties from EES or other source. Get 12.50 exercise solution
12–51E Reconsider Prob. 12–50E. Using EES (or other) software, plot the Joule-Thomson coefficient for nitrogen over the pressure range 100 to 1500 psia at the enthalpy values 100,175,and 225 Btu/lbm. Discuss the results. Get 12.51 exercise solution
12–52 Estimate the Joule-Thomson coefficient of refrigerant-134a at 0.7 MPa and 50°C. Get 12.52 exercise solution
12–53 Steam is throttled slightly from 1 MPa and 300°C. Will the temperature of the steam increase, decrease, or remain the same during this process? Get 12.53 exercise solution
12–54C What is the enthalpy departure? Get 12.54 exercise solution
12–55C On the generalized enthalpy departure chart, the normalized enthalpy departure values seem to approach zero as the reduced pressure PR approaches zero. How do you explain this behavior? Get 12.55 exercise solution
12–56C Why is the generalized enthalpy departure chart prepared by using PR and TR as the parameters instead of P and T? Get 12.56 exercise solution
12–57 Determine the enthalpy of nitrogen, in kJ/kg, at 175 K and 8 MPa using (a) data from the ideal-gas nitrogen table and (b) the generalized enthalpy departure chart. Compare your results to the actual value of 125.5 kJ/kg. Get 12.57 exercise solution
12–58E Determine the enthalpy of nitrogen, in Btu/lbm, at 400 R and 2000 psia using (a) data from the ideal-gas nitrogen table and (b) the generalized enthalpy chart. Compare your results to the actual value of 177.8 Btu/lbm. Get 12.58 exercise solution
12–59 What is the error involved in the (a) enthalpy and (b) internal energy of CO2 at 350 K and 10 MPa if it is assumed to be an ideal gas? Answers: (a) 50%, (b) 49% Get 12.59 exercise solution
12–60 Determine the enthalpy change and the entropy change of nitrogen per unit mole as it undergoes a change of state from 225 K and 6 MPa to 320 K and 12 MPa, (a) by assuming ideal-gas behavior and (b) by accounting for the deviation from ideal-gas behavior through the use of generalized charts. Get 12.60 exercise solution
12–61 Determine the enthalpy change and the entropy change of CO2 per unit mass as it undergoes a change of state from 250 K and 7 MPa to 280 K and 12 MPa, (a) by assuming ideal-gas behavior and (b) by accounting for the deviation from ideal-gas behavior. Get 12.61 exercise solution
12–62 Methane is compressed adiabatically by a steady-flow compressor from 2 MPa and -10°C to 10 MPa and 110°C at a rate of 0.55 kg/s. Using the generalized charts, determine the required power input to the compressor. Get 12.62 exercise solution
12–63 Propane is compressed isothermally by a piston– cylinder device from 100°C and 1 MPa to 4 MPa. Using the generalized charts, determine the work done and the heat transfer per unit mass of propane. Get 12.63 exercise solution
12–64 Reconsider Prob. 12–63. Using EES (or other) software, extend the problem to compare the solutions based on the ideal-gas assumption, generalized chart data, and real fluid data. Also extend the solution to methane. Get 12.64 exercise solution
12–65E Propane is compressed isothermally by a piston– cylinder device from 200°F and 200 psia to 800 psia. Using the generalized charts, determine the work done and the heat transfer per unit mass of the propane. Get 12.65 exercise solution
12–66 Determine the exergy destruction associated with the process described in Prob. 12–63. Assume T0 = 30°C. Get 12.66 exercise solution
12–67 Carbon dioxide enters an adiabatic nozzle at 8 MPa and 450 K with a low velocity and leaves at 2 MPa and 350 K. Using the generalized enthalpy departure chart, determine the exit velocity of the carbon dioxide. Get 12.67 exercise solution
12–68 Reconsider Prob. 12–67. Using EES (or other) software, compare the exit velocity to the nozzle assuming ideal-gas behavior, the generalized chart data, and EES data for carbon dioxide. Get 12.68 exercise solution
12–69 A 0.08-m3 well-insulated rigid tank contains oxygen at 220 K and 10 MPa. A paddle wheel placed in the tank is turned on, and the temperature of the oxygen rises to 250 K. Using the generalized charts, determine (a) the final pressure in the tank and (b) the paddle-wheel work done during this process. Get 12.69 exercise solution
12–70 Carbon dioxide is contained in a constant-volume tank and is heated from 100°C and 1 MPa to 8 MPa. Determine the heat transfer and entropy change per unit mass of the carbon dioxide using (a) the ideal-gas assumption, (b) the generalized charts, and (c) real fluid data from EES or other sources. Get 12.70 exercise solution
12–71 For B>= 0, prove that at every point of a singlephase region of an h-s diagram, the slope of a constantpressure (P = constant) line is greater than the slope of a constant-temperature (T = constant) line, but less than the slope of a constant-volume (v = constant) line. Get 12.71 exercise solution
12–72 Using the cyclic relation and the first Maxwell relation, derive the other three Maxwell relations. Get 12.72 exercise solution
12–73 Starting with the relation dh = T ds + v dP, show that the slope of a constant-pressure line on an h-s diagram (a) is constant in the saturation region and (b) increases with temperature in the superheated region. Get 12.73 exercise solution
12–74 Derive relations for (a) du,( b) dh, and (c) ds of a gas that obeys the equation of state (P + a/v2)v = RT for an isothermal process. Get 12.74 exercise solution
12–75 Show that
Get 12.75 exercise solution
12–76 Estimate the cp of nitrogen at 300 kPa and 400 K, using (a) the relation in the above problem and (b) its definition. Compare your results to the value listed in Table A–2b. Get 12.76 exercise solution
12–77 Steam is throttled from 4.5 MPa and 300°C to 2.5 MPa. Estimate the temperature change of the steam during this process and the average Joule-Thomson coefficient. Get 12.77 exercise solution
12–78 A rigid tank contains 1.2 m3 of argon at -100°C and 1 MPa. Heat is now transferred to argon until the temperature in the tank rises to 0°C. Using the generalized charts, determine (a) the mass of the argon in the tank, (b) the final pressure, and (c) the heat transfer. Answers: (a) 35.1 kg, (b) 1531 kPa, (c) 1251 kJ Get 12.78 exercise solution
12–79 Argon gas enters a turbine at 7 MPa and 600 K with a velocity of 100 m/s and leaves at 1 MPa and 280 K with a velocity of 150 m/s at a rate of 5 kg/s. Heat is being lost to the surroundings at 25°C at a rate of 60 kW. Using the generalized charts, determine (a) the power output of the turbine and (b) the exergy destruction associated with the process. Get 12.79 exercise solution
12–80 Reconsider Prob. 12–79. Using EES (or other) software, solve the problem assuming steam is the working fluid by using the generalized chart method and EES data for steam. Plot the power output and the exergy destruction rate for these two calculation methods against the turbine exit pressure as it varies over the range 0.1 to 1 MPa when the turbine exit temperature is 455 K. Get 12.80 exercise solution
12–81E Argon gas enters a turbine at 1000 psia and 1000 R with a velocity of 300 ft/s and leaves at 150 psia and 500 R with a velocity of 450 ft/s at a rate of 12 lbm/s. Heat is being lost to the surroundings at 75°F at a rate of 80 Btu/s. Using the generalized charts, determine (a) the power output of the turbine and (b) the exergy destruction associated with the process. Answers: (a) 922 hp, (b) 121.5 Btu/s Get 12.81 exercise solution
12–82 An adiabatic 0.2-m3 storage tank that is initially evacuated is connected to a supply line that carries nitrogen at 225 K and 10 MPa. A valve is opened, and nitrogen flows into the tank from the supply line. The valve is closed when the pressure in the tank reaches 10 MPa. Determine the final temperature in the tank (a) treating nitrogen as an ideal gas and (b) using generalized charts. Compare your results to the actual value of 293 K. Get 12.82 exercise solution
12–83 For a homogeneous (single-phase) simple pure substance, the pressure and temperature are independent properties, and any property can be expressed as a function of these two properties. Taking v = v(P, T), show that the change in specific volume can be expressed in terms of the volume expansivity b and isothermal compressibility a as
Also, assuming constant average values for b and a, obtain a relation for the ratio of the specific volumes v2/v1 as a homogeneous system undergoes a process from state 1 to state 2. Get 12.83 exercise solution
12–84 Repeat Prob. 12–83 for an isobaric process. Get 12.84 exercise solution
12–85 The volume expansivity of water at 20°C is b = 0.207 x 10-6 K-1. Treating this value as a constant, determine the change in volume of 1 m3 of water as it is heated from 10°C to 30°C at constant pressure. Get 12.85 exercise solution
12–86 The volume expansivity b values of copper at 300 K and 500 K are 49.2 x 10-6 K-1 and 54.2 x 10-6 K-1, respectively, and b varies almost linearly in this temperature range. Determine the percent change in the volume of a copper block as it is heated from 300 K to 500 K at atmospheric pressure. Get 12.86 exercise solution
12–87 Starting with mJT = (1/cp) [T(dv/dT)p - v] and noting that Pv = ZRT, where Z = Z(P, T) is the compressibility factor, show that the position of the Joule-Thomson coefficient inversion curve on the T-P plane is given by the equation (dZ/dT)P = 0. Get 12.87 exercise solution
12–88 Consider an infinitesimal reversible adiabatic compression or expansion process. By taking s = s(P, v) and using the Maxwell relations, show that for this process Pvk = constant, where k is the isentropic expansion exponent defined as
Also, show that the isentropic expansion exponent k reduces to the specific heat ratio cp/cv for an ideal gas. Get 12.88 exercise solution
12–89 Refrigerant-134a undergoes an isothermal process at 60°C from 3 to 0.1 MPa in a closed system. Determine the work done by the refrigerant-134a by using the tabular (EES) data and the generalized charts, in kJ/kg. Get 12.89 exercise solution
12–90 Methane is contained in a piston–cylinder device and is heated at constant pressure of 4 MPa from 100 to 350°C. Determine the heat transfer, work and entropy change per unit mass of the methane using (a) the ideal-gas assumption, (b) the generalized charts, and (c) real fluid data from EES or other sources. Get 12.90 exercise solution
12–91 A substance whose Joule-Thomson coefficient is negative is throttled to a lower pressure. During this process, (select the correct statement) (a) the temperature of the substance will increase. (b) the temperature of the substance will decrease. (c) the entropy of the substance will remain constant. (d) the entropy of the substance will decrease. (e) the enthalpy of the substance will decrease. Get 12.91 exercise solution
12–92 Consider the liquid–vapor saturation curve of a pure substance on the P-T diagram. The magnitude of the slope of the tangent line to this curve at a temperature T (in Kelvin) is (a) proportional to the enthalpy of vaporization hfg at that temperature. (b) proportional to the temperature T. (c) proportional to the square of the temperature T. (d) proportional to the volume change vfg at that temperature. (e) inversely proportional to the entropy change sfg at that temperature. Get 12.92 exercise solution
12–93 Based on the generalized charts, the error involved in the enthalpy of CO2 at 350 K and 8 MPa if it is assumed to be an ideal gas is (a) 0 (b) 20% (c) 35% (d) 26% (e) 65% Get 12.93 exercise solution
12–94 Based on data from the refrigerant-134a tables, the Joule-Thompson coefficient of refrigerant-134a at 0.8 MPa and 100°C is approximately (a) 0 ( b) -5°C/MPa (c) 11°C/MPa (d)8°C/MPa (e) 26°C/MPa Get 12.94 exercise solution
12–95 For a gas whose equation of state is P(v - b) = RT, the specified heat difference cp -cv is equal to (a) R (b) R - b (c) R + b (d) 0 (e) R(1 + v/b) Get 12.95 exercise solution
