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1 CBE 141: Chemical Engineering Thermodynamics, Spring 2017, UC Berkeley Midterm 2 FORM B March 23, 2017 Time: 80 minutes, closed-book and closed-notes, one-sided 8 ½ x 11 equation sheet allowed lease 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. oint Totals: roblem roblem 5 15 roblem 6 15 roblem 7 15 roblem 8 20 roblem 9 20 roblem roblem roblem Total 185 1

2 roblem 1 (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 roblem 2 (5 points) ure 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, v = 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. ressure will change so that chemical and mechanical forces balance D. Nothing will happen roblem 3 (5 points) Consider a distillation (i.e. separation) of a heptane (boiling point 209 C)/decane (boiling point 345 C) mixture performed at constant pressure with multiple separation columns connected in parallel. There is only one inlet in the first column, and one outlet in the last column. Which of the following are true (choose one). A. The volume of the liquid product at the outlet will increase if you increase the number of columns operated in parallel. B. The purity of the final separated product will decrease if you increase the number of columns operated in parallel. 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 roblem 4 (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 oynting factor assumes that a fluid is incompressible. D. The Clausius-Clapeyron equation assumes that the vapor pressure of a gas is independent of temperature. 2

3 roblem 5: (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? roblem 6 (15 points) a) 7 oints Show that ( T V ) S = ( S ) V du = TdS dv T = ( U S ) V = ( U V ) S ( T V ) = ( S V ( U S ) ) = ( V s S ( U V ) ) = ( S V S ) V 3

4 b) 8 points 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 ) S = ( V S ) Enthalpy: dh = TdS + Vd ( S V ) = ( T ) T V Helmholtz Free Energy: da = SdT dv roblem 7 (15 points) 3 points each For each of the statements below, mark whether they are true (T) or false (F): 1) F For a system in which temperature and volume are held constant, equilibrium is reached when Gibbs free energy is minimized. 2) T An adiabatic throttle is always an irreversible process. 3) T As a fluid approaches ideal behavior, the fugacity coefficient approaches 1. 4) F A refrigerator moves heat from a hot place to a cold place, extracting work in the process. 5) F Fugacity is an extensive property. 4

5 roblem 8 (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) (3 points) Label the y-axis with one of the following: [Enthalpy, Temperature, ressure, Entropy, Miscibility] b) (4 points) Label the dew line (point where condensation begins) and the bubble line (where liquid boiling begins) c) (3 points) The liquid at point A contains more decane (decane, heptane, or equal) than the liquid at point B? d) (3 points) The liquid at point A boils at a higher (higher, lower, or same) temperature than the liquid at point B? e) (4 points) Draw the bubble line and the dew line on the graph below if we plot the molar composition (x-axis) as a function of decane mole fraction instead of heptane mole fraction. f) (3 points) The area within the envelope (between dew and bubble lines) would decrease [increase, decrease, stay the same] if we replace decane (boiling point 345 C) with octane (boiling point 257 C). 5

6 roblem 9 (20 points) a) From the following definitions of residuals: 0 H R = 2 ( Z d ) T 0 S R = ( Z d ) T 0 R (Z 1) d What is the expression for residual Gibbs free energy, G R, as a function of R,T,, and Z? Answer: G R = H R TS R 0 G R = (Z 1) d b) Find the residual Gibbs free energy for a fluid at pressure, that obeys the truncated virial equation of state, which is Z = 1 + B (T) Answer: G R = (Z 1) d 0 G R = (B (T)) d 0 G R = B (T) d 0 G R = B (T) 6

7 roblem 10 (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) (8 points) On the T-S plot below, draw a vapor-compression refrigeration cycle b) (4 points) Using arrows, label on the diagram the directionality of the cycle if you wanted to move heat from low to high temperature reservoirs c) (4 points) 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) (4 points) 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) (4 points) For each segment of the cycle, state which piece of equipment is used f) (6 points) Running the vapor-compression cycle relies on the Joule-Thompson Effect. The Joule-Thompson coefficient is defined as: μ JT = ( T ) 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? This is relevant for the throttle. The coefficient is positive (heating). If it was negative, your working fluid would warm upon throttling and the refrigeration cycle would not work or be extremely inefficient T S 7

8 T Isobaric/ condenser Isenthalpic/ throttle Isentropic/ compressor Isobaric Or isothermal/ evaporator S 8

9 roblem 11 (20 points) a) 10 points Show that the definition for fugacity f(, T) e G R(,T) leads to f( 2, T) f( 1, T) = eg( 2,T) G( 1,T) where G R is the residual Gibbs free energy, G(, T) is the total Gibbs free energy at pressure and temperature T. The subscripts 1 and 2 denote an initial and final states respectively G = G R + G IG f 1 = 1 e G R ( 1 ) f 2 = 2 e G R( 2) G R( 1) f 1 1 G IG ( 2 ) G IG ( 1 ) = ln ( 2 ) 1 2 = e G IG( 2) G IG( 1) 1 f 2 = e G IG( 2) G IG( 1) e G R ( 1 ) G R( 2 ) = e G( 2,T) G( 1,T) f 1 b) 10 points Estimate the fugacity coefficient (φ = f/ ) of methane at = 5.0 Ma and T = 273 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) (Ma) V (L/mol) H (kj/mol) S (J/mol*K) Note: R=8.31 J K -1 mol -1 f 2 = e G( 2,T) G( 1,T) f 1 Limit at 1 0, f 1 = 1 ΔG = Vd or ΔG = (H TS) 2 (H TS) 1 f 2 = 1 e G( (H TS) 2,T) G(1,T) 2 (H TS) 1 = 1 e 6240 ( 19900) f 2 = 0.01 exp ( ) = φ = 4.07 = a. Because φ 1, methane is nonideal at these conditions. And b. because φ < 1, we can interpret that molecules are attracted to each other. 9

10 roblem 12 (30 points) a) Calculate the ideal work per mole needed to compress air at 50 C and 10 Ka to 60 Ka. 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) Answer: b) c) H 1 = kj mol S 1 = J mol S 2 = J mol and 2 = 60 ka H 2 = kj mol T = 225 o C W = ΔH W ideal = H 2 H 1 = = kj mol W actual = W ideal η η = W ideal W actual = = kj mol W actual = ΔH actual = H 2 H 1 H 2 = H 1 + ΔH = = kj mol Temperature (C) ressure (ka) Volume (l/mol) Internal Energy (kj/mol) Enthalpy (kj/mol) Entropy (J/mol*K) Cv (J/mol*K) Cp (J/mol*K) liquid liquid liquid liquid liquid liquid liquid vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor hase 10

11 vapor Temperature (C) ressure (ka) Volume (l/mol) Internal Energy (kj/mol) Enthalpy (kj/mol) Entropy (J/mol*K) Cv (J/mol*K) Cp (J/mol*K) vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor vapor Additional space provided for work. hase 11

12 Additional space provided for work. 12