# TAMU CHEN 205 - extra-sol234 (6 pages)

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- School:
- Texas A&M University
- Course:
- Chen 205 - Chem Engr Thermo I

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ME 24 221 THERMODYNAMICS I Solutions to extra problems in Chapters 2 3 and 4 December 5 2000 J Murthy 2 25 Two reservoirs A and B open to the atmosphere are connected with a mercury manometer Reservoir A is moved up down so the two top surfaces are level at h3 as shown in Fig P2 25 Assuming that you know A Hg and measure the heights h1 h2 and h3 find the density B Solution Balance forces on each side P0 Ag h3 h2 Hggh2 P0 Bg h3 h1 Hggh1 h 3 h2 h 2 h1 B A Hg h 3 h1 h 3 h1 2 32 Two piston cylinder arrangements A and B have their gas chambers connected by a pipe Cross sectional areas are AA 75 cm2 and AB 25 cm2 with the piston mass in A being mA 25 kg Outside pressure is 100 kPa and standard gravitation Find the mass mB so that none of the pistons have to rest on the bottom Solution P0 P0 Force balance for both pistons A mPAg P0AA PAA B A B F F mPBg P0AB PAB Same P in A and B gives no flow between them mPBg mPAg P0 P0 AA AB mPB mPA AA AB 25 25 75 8 33 kg 2 36 Two cylinders are connected by a piston as shown in Fig P2 36 Cylinder A is used as a hydraulic lift and pumped up to 500 kPa The piston mass is 25 kg and there is standard gravity What is the gas pressure in cylinder B Solution Force balance for the piston PBAB mpg P0 AA AB PAAA AA 4 0 12 0 00785 m2 AB 4 0 0252 0 000491 m2 PBAB PAAA mpg P0 AA AB 500 0 00785 25 9 807 1000 100 0 00785 0 000491 2 944 kN PB 2 944 0 000491 5996 kPa 6 0 MPa 4 21 A cylinder having an initial volume of 3 m3 contains 0 1 kg of water at 40 C The water is then compressed in an isothermal quasi equilibrium process until it has a quality of 50 Calculate the work done in the process Assume the water vapor is an ideal gas Solution C V Water v1 V1 m T PG 40 oC 3 P1 Tbl B 1 1 PG 7 384 kPa very low so H2O ideal gas from 1 2 1 2 P 1 PG 2 PdV P1V1ln W12 1 vG v1 7 384 19 52 4 8 kPa 30 V2 mv2 0 1 19 52 1 952 m3 v T C 3 30 m3 kg vG 0 1 V2 V1 4 8 3 0 ln 1 952 6 19 kJ 3 v3 0 001008 0 5 19 519 9 7605 V3 mv3 0 976 m3 3 P C P g W23 PdV Pg V3 V2 7 384 0 976 1 952 7 21 kJ 2 Total work W13 6 19 7 21 13 4 kJ 3 12 Air in a tank is at 1 MPa and room temperature of 20 C It is used to fill an initially empty balloon to a pressure of 200 kPa at which point the diameter is 2 m and the temperature is 20 C Assume the pressure in the balloon is linearly proportional to its diameter and that the air in the tank also remains at 20 C throughout the process Find the mass of air in the balloon and the minimum required volume of the tank Solution Assume air is an ideal gas V2 4 3 r3 4 3 23 33 51 m3 Balloon final state m2bal P2 V2 RT2 200 33 51 0 287 293 15 79 66 kg P2 200 kPa m2 tank P2 VTANK RT2 Initial mass must be enough m1 m2bal m2 tank P1V1 R T1 Tank must have P1VTANK R T1 m2bal P2VTANK RT2 VTANK RTm2bal P1 P2 0 287 293 15 79 66 1000 200 8 377 m 3 22 3 Is it reasonable to assume that at the given states the substance behaves as an ideal gas Solution a Oxygen O2 at 30 C 3 MPa Ideal Gas T Tc 155 K from A 2 b Methane CH4 at 30 C 3 MPa Ideal Gas T Tc 190 K from A 2 c Water H2O NO compressed liquid P Psat B 1 1 at 30 C 3 MPa d R 134a at 30 C 3 MPa NO compressed liquid P Psat B 5 1 e R 134a at 30 C 100 kPa Ideal Gas P is low Psat B 5 1 ln P c d Cr P Liq a b e Vapor T 3 35 Determine the mass of methane gas stored in a 2 m3 tank at 30 C 3 MPa Estimate the percent error in the mass determination if the ideal gas model is used Solution The methane Table B 7 2 linear interpolation between 225 and 250 K v 0 03333 243 15 225 0 03896 0 03333 0 03742 m3 kg 250 225 m V v 2 0 03742 53 45 kg Ideal gas assumption v RT P 0 51835 243 15 3000 0 042 m V v 2 0 042 47 62 kg Error 5 83 kg 10 9 too small 4 10 A piston cylinder contains 1 kg of liquid water at 20 C and 300 kPa Initially the piston floats similar to the setup in Problem 4 7 with a maximum enclosed volume of 0 002 m3 if the piston touches the stops Now heat is added so a final pressure of 600 kPa is reached Find the final volume and the work in the process Solution Take CV as the water which is a control mass m2 m1 m Table B 1 1 20 C Psat 2 34 kPa State 1 Compressed liquid P 2 P1 1 v vf 20 0 001002 m3 kg State 1a vstop 0 002 m3 kg 300 kPa 1a V V stop State 2 Since P 600 Plift then v vstop 0 002 and V 0 002 m3 For the given P vf v v g so 2 phase T Tsat 158 85 C Work is done while piston moves at Plift constant 300 kPa so we get 1W2 P dV m Plift v2 v1 1 300 0 002 0 001002 0 30 kJ 4 13 Air in a spring loaded piston cylinder has a pressure that is linear with volume P A BV With an initial state of P 150 kPa V 1 L and a final state of 800 kPa and volume 1 5 L it is similar to the setup in Problem 3 16 Find the work done by the air Solution Knowing the process equation P A BV giving a linear variation of pressure versus volume the straight line in the P V diagram is fixed by the two points as state 1 and state 2 The work as the integral of PdV equals the area under the process curve in the P V diagram State 1 P1 150 kPa P 2 1 V State 2 P2 800 kPa V2 1 5 L 0 …

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