CBE 141: Chemical Engineering Thermodynamics, Spring 2017, UC Berkeley Midterm 2 FORM A March 23, 2017 Time: 80 minutes, closed-book and closed-notes, one-sided 8 ½ x 11 equation sheet allowed Please show all work and clearly mark your answers. Write your name on any additional pages of scratch work. You must turn in your equation sheet with your exam. Point Totals: Problem 1-4 20 Problem 5 15 Problem 6 15 Problem 7 15 Problem 8 20 Problem 9 20 Problem 10 30 Problem 11 20 Problem 12 30 Total 185 1
Problem 1 (5 points) Consider a distillation (i.e. separation) of a heptane (boiling point 98.42 C)/decane (boiling point 174.1 C) mixture performed at constant pressure with multiple separation columns connected in series. There is only one inlet in the first column, and one outlet in the last column. Which of the following is always true? (Choose one) A. The volume of the liquid product at the outlet will increase if you increase the number of columns operated in series. B. The purity of the final separated product will decrease if you increase the number of columns operated in series. C. The vapor is enriched in the more volatile component, and the liquid is enriched in the less volatile component. D. The liquid is enriched in the more volatile component, and the vapor is enriched in the less volatile component. Problem 2 (5 points) Which of the following is NOT true? (Choose one) A. Multiple phases at the same temperature and pressure are in equilibrium when the chemical potential of each species is the same in all phases. B. For a pure species at equilibrium, the fugacities of the vapor and liquid phases are equal. C. The Poynting factor assumes that a fluid is incompressible. D. The Clausius-Clapeyron equation assumes that the vapor pressure of a gas is independent of temperature. Problem 3 (5 points) Pure species A exists as a mixture of liquid and vapor. The pressure for the liquid and vapor are the same (i.e. they are in mechanical equilibrium, P v = P l ), but the fugacity of the liquid is higher than the fugacity of the vapor (f l > f v ). What will happen? (Choose one) A. The amount of species A in the vapor phase will increase and the amount in liquid phase will decrease B. The amount of species A in the liquid phase will increase and the amount in the vapor phase will decrease C. Pressure will change so that chemical and mechanical forces balance D. Nothing will happen Problem 4 (5 points) For a refrigerator, the following are true (you may circle more than one answer): A) The coefficient of performance is larger than unity when 2Q absorbed>q rejected B) The coefficient of performance is always finite C) Q rejected is always larger than Q absorbed D) The work required to run a refrigeration cycle is always larger than zero E) The working fluid in a refrigeration cycle must have a low heat capacity 2
Problem 5: (15 points) For each of the statements below, mark whether they are true (T) or false (F): 1) For a system in which temperature and volume are held constant, equilibrium is reached when Gibbs free energy is minimized. 2) An adiabatic throttle is always an irreversible process. 3) As a fluid approaches ideal behavior, the fugacity coefficient approaches 1. 4) A refrigerator moves heat from a hot place to a cold place, extracting work in the process. 5) Fugacity is an extensive property. Problem 6 (15 points) a) Show that ( T V ) S = ( P S ) V b) Fill out the Maxwell relations below and state which Legendre transform of internal energy they can be derived from (hint: which combined first and second laws can represent these energy functions?): ( T P ) = S ( S V ) = T 3
Problem 7 (15 points) A tray of ice cubes is placed in a freezer having an ideal coefficient of performance of 10. If the room temperature is 32 C, will the ice cubes melt or remain frozen? 4
Problem 8 (20 points) a) Show that the definition for fugacity leads to f(p, T) Pe G R(P,T) RT f(p 2, T) f(p 1, T) = eg(p 2,T) G(P 1,T) RT where G R is the residual Gibbs free energy, G(P, T) is the total Gibbs free energy at pressure P and temperature T. The subscripts 1 and 2 denote an initial and final states respectively b) Estimate the fugacity coefficient (φ = f/p ) of methane at P = 5.0 MPa and T = 273 K AND, comment on what this result says about the interactions between methane molecules at these conditions (i.e. ideal? Non-ideal?). Tabulated values for methane at various pressures is given below. T (K) P (MPa) V (L/mol) H (kj/mol) S (J/mol*K) 273 0.01 227 13.7 123 273 2.0 1.08 13.4 78.2 273 5.0 0.404 12.6 69.0 Note: R=8.31 J K -1 mol -1 5
Problem 9 (20 points) a) From the following definitions of residuals: P 0 H R = RT 2 ( Z dp ) T P P P 0 S R = RT ( Z dp ) T P P P 0 R (Z 1) dp P What is the expression for residual Gibbs free energy, G R, as a function of R,T,P, and Z? b) Find the residual Gibbs free energy for a fluid at pressure P, that obeys the truncated virial equation of state, which is Z = 1 + B (T)P 6
Problem 10 (30 points) a) Calculate the ideal work per mole needed to compress steam at 50 C and 10 KPa to 60 KPa. What is the temperature of the exiting stream? b) For a compressor that is 67% efficient, what is the actual work required for this process? c) What is the enthalpy of the exiting stream for the real process? You may use the following tabulated data (note, not all is needed) Temperature (C) Pressure (kpa) Volume (l/mol) Internal Energy (kj/mol) Enthalpy (kj/mol) Entropy (J/mol*K) Cv (J/mol*K) Cp (J/mol*K) 20 10 0.018048 1.5117 1.5119 5.3412 74.89 75.382 liquid 25 10 0.018069 1.8885 1.8886 6.6156 74.545 75.332 liquid 30 10 0.018095 2.265 2.2652 7.8682 74.178 75.305 liquid 35 10 0.018124 2.6415 2.6417 9.1 73.792 75.295 liquid 40 10 0.018157 3.018 3.0182 10.312 73.388 75.297 liquid 45 10 0.018194 3.3945 3.3947 11.505 72.969 75.31 liquid 45.806 10 0.0182 3.4553 3.4554 11.695 72.9 75.313 liquid 45.806 10 264.29 43.906 46.549 146.8 26.329 34.95 vapor 50 10 267.83 44.017 46.695 147.26 26.153 34.733 vapor 55 10 272.05 44.148 46.868 147.79 26.035 34.58 vapor 60 10 276.26 44.278 47.041 148.31 25.968 34.486 vapor 65 10 280.47 44.408 47.213 148.83 25.928 34.424 vapor 70 10 284.67 44.538 47.385 149.33 25.904 34.382 vapor 75 10 288.86 44.668 47.557 149.83 25.891 34.354 vapor 80 10 293.06 44.798 47.729 150.32 25.887 34.336 vapor 85 10 297.25 44.928 47.9 150.8 25.889 34.326 vapor 90 10 301.43 45.058 48.072 151.28 25.896 34.323 vapor 95 10 305.62 45.187 48.244 151.75 25.908 34.326 vapor 100 10 309.8 45.317 48.415 152.21 25.924 34.333 vapor Phase Temperature (C) Pressure (kpa) Volume (l/mol) Internal Energy (kj/mol) Enthalpy (kj/mol) Entropy (J/mol*K) Cv (J/mol*K) Cp (J/mol*K) 220 60 68.16 48.454 52.543 146.93 27.007 35.454 vapor 225 60 68.858 48.589 52.721 147.26 27.05 35.49 vapor 230 60 69.556 48.725 52.898 147.64 27.093 35.527 vapor 235 60 70.254 48.861 53.076 147.99 27.137 35.566 vapor 240 60 70.951 48.997 53.254 148.34 27.182 35.607 vapor 245 60 71.648 49.133 53.432 148.68 27.228 35.648 vapor 250 60 72.346 49.27 53.61 149.03 27.275 35.69 vapor 255 60 73.043 49.406 53.789 149.37 27.323 35.734 vapor Phase 7
260 60 73.739 49.543 53.968 149.7 27.371 35.778 vapor 265 60 74.436 49.68 54.147 150.04 27.42 35.824 vapor 270 60 75.133 49.818 54.326 150.37 27.469 35.87 vapor 275 60 75.829 49.956 54.505 150.7 27.52 35.917 vapor 280 60 76.525 50.093 54.685 151.02 27.57 35.964 vapor 285 60 77.221 50.232 54.865 151.35 27.621 36.012 vapor 290 60 77.917 50.37 55.045 151.67 27.673 36.061 vapor 295 60 78.613 50.509 55.226 151.99 27.725 36.111 vapor 300 60 79.309 50.648 55.406 152.3 27.778 36.161 vapor 305 0.06 80.005 50.787 55.587 152.62 27.83 36.211 vapor 310 0.06 80.7 50.926 55.768 152.93 27.884 36.262 vapor 315 0.06 81.396 51.066 55.95 153.24 27.937 36.313 vapor 320 0.06 82.091 51.206 56.131 153.55 27.991 36.365 vapor Space provided for work. 8
Problem 11 (20 points) Below is a phase diagram for a mixture of heptane (more volatile) and decane (less volatile). The graph is plotted as a mole fraction of heptane. a) Label the y-axis with one of the following: [Enthalpy, Temperature, Pressure, Entropy, Miscibility] b) Label the dew line (point where condensation begins) and the bubble line (where liquid boiling begins) c) The liquid equilibrated from point A contains more (decane, heptane, or equal) than the liquid equilibrated from point B? d) The liquid equilibrated from point A boils at a (higher, lower, or same) temperature than the liquid equilibrated from point B? e) Draw the bubble line and the dew line on the graph below if we plot the molar composition (xaxis) as a function of decane mole fraction instead of heptane mole fraction. f) The area within the envelope (between dew and bubble lines) would [increase, decrease, stay the same] if we replace decane (boiling point 174.1 C) with octane (boiling point 125 C). 9
Problem 12 (30 points) You are asked to build a vapor-compression refrigeration cycle. At your disposal, you have the following pieces of equipment: Condenser Evaporator Throttle Compressor a) On the T-S plot below, draw a vapor-compression refrigeration cycle b) Using arrows, label on the diagram the directionality of the cycle if you wanted to move heat from low to high temperature reservoirs c) For each segment of the cycle, label with one of the following: isobaric ; isochoric ; isothermal ; isentropic ; isenthalpic; athermal (each may be used more than once, or not at all) d) Draw an arrow labeled W for the segment of the cycle where work is either input into the cycle or where work is extracted from the cycle. Also: specify whether work is input into or extracted from the cycle at this segment. e) For each segment of the cycle, state which piece of equipment is used f) Running the vapor-compression cycle relies on the Joule-Thompson Effect. The Joule- Thompson coefficient is defined as: μ JT = ( T P ) H What step of the vapor-compression refrigeration cycle is this coefficient relevant for? What is the sign of the coefficient at this step? What would happen if the coefficient had the opposite sign? 10
Additional space provided for work. 11
Additional space provided for work. 12
Additional space provided for work. 13