A stirred tank with volume Vt(L) is charged with V](L) of

Chapter 8, Problem 8.94

(choose chapter or problem)

A stirred tank with volume Vt(L) is charged with V](L) of a liquid, B. The space above the liquid (volume Vg = Vt - VI) is filled with a pure gas, A, at an initial pressure Po(atm). The initial system temperature is To(K). The stirrer in the tank is turned on, and A begins to dissolve in B. The dissolution continues until the liquid is saturated with A at the final system temperature (T) and pressure (P). The equilibrium solubility of A in B is governed by the following expression, which relates the molar A/B ratio in the liquid to the partial pressure of A in the gas phase (which in turn equals the pressure in the tank, since the gas is pure A): r(mol A/mol B) = ksPA(atm) where ks[mol A/(mol Batm)] = Co + c1T(K) *Computer problems. 437 When solving the problems to be given, use the following variable definitions: M A , MB = molecular weights of A and B CvA, CvB , Cvs[J/(g'K)] =constant-volume heat capacities of A(g), B(I), and solutions of A in B. respectively 5GB = specific gravity of B(l) 6.Us (llmol A dissolved) = internal energy ofsolution at 298 K (independent of composition over the range of concentrations to be considered) nAi), nso = g-moles of A(g) and B(l) initially charged into the tank nA(I), nA(v) = g-moles of A dissolved and remaining in the gas phase at equilibrium. respectively Make the following assumptions: A negligible amount of B evaporates. The tank is adiabatic and the work input to the tank from the stirrer is negligible. The gas phase behaves ideally. The volumes of the liquid and gas phases may be considered constant. The heat capacities CvA, Cvs, and Cvs are constant. independent of temperature and (in the case of C,JS ) solution composition. (a) Use material balances, the given equilibrium solubility relation. and the ideal gas equation of state to derive expressions for nAO, nso, nA(v), n-\(I). and P in terms of the final temperarure. T. and variables MA, MB, 5GB, Vt , VI, To. Po, Co, and Cl. Then use an energy balance to deri\'e the following equation: T = 298 + nAO)(-t:.Us) + (nAoMAC,'A + nBoMBC,s)(To - 298) nA(v)M.-\CvA + (nA(J)MA(l) + nSMB)C,-s (b) Write a spreadsheet to calculate T from specified values of MA ( = -+7)..HB( = 26). 5GB( = 1.-6). Vt ( = 20.0), VI( = 3.0), co( = 1.54 x 10-3 ), Cl (= -1.60 x 10-6 ). CrA( = 0.831). C B( = :.S51. Cvs( = 3.80), and t:.Us( = -1.74 x 105 ). and a number of different values of Tn and Pl. The spreadsheet should have the structure given below. (Calculated values are shown for one initial temperature and pressure.) i 8.94 I I I I i I i I I I i I I MB I ! 5GB I i Vt MA CvA CvB i cO cl Des CYS I I 20.0 26.0 I i 47.0 0.831 3.85 I 1.76 0.0015-+ -1.60E-06 -17-+000 .3.80 i i I I I I ! i I i I VI 3.0 3.0 TO 300 300 PO 1.0 5.0 Vg nBO nAO T nA(v) nA(I) P Tw.k i I 3.0 300 10.0 17.0 203.1 6.906 I 320.0 5.222 1.684 S.l 31-k2 I I i 3.0 300 I 20.0 ! 3.0 330 I l.0 I I I I I I I 3.0 330 5.0 I i 3.0 I 330 I 10.0 I ; I : I i 1 I 3.0 330 I 20.0 i The values of Vg, nso, and nAO should first be calculated from the given values of the other variables. Next, a value of T should be guessed (in the example in the table. the guessed \";llue 438 Chapter 8 Balances on Nonreactive Processes is 320 K), the values of nA(v), nA(l), and P should be calculated from the equations derived in part (a), and the temperature should be recalculated from the energy balance in the column labeled Teale (it equals 314.2 in the example). The value of T should then be varied until it equals the recalculated value of Teale. (Suggestion: Create a new cell as T - Teale and use goalseek to find the value of T that drives T - Tcalc to zero.) Enter the formulas in the cells for VI = 3.0 L, To = 300 K, and Po = 10.0 atm, and verify that your cell values match those shown above. Then find the correct value of Tusing the procedure just described, copy the formulas into the other rows of the table, and determine T fur each set of initial conditions. Summarize the effects of the initial temperature and pressure on the adiabatic temperature rise and briefly explain why your results make sense. (c) Write a computer program to perform the same calculations done with the spreadsheet in part (b). Define values of Vb MA , CvA , MB , CvB , SGB, Co, Cl, flOs, and Cvs ' Use the values shown in the fourth row of the spreadsheet. Read in a set of values of VB, To, and Po. Have the program terminate if VB :s 0.0. If a positive value is read in for VB, calculate Vo , nB, and nAO' Assume a value of T. (Try LITo as a first guess.) Calculate nAry), nA(l), and P from the equations derived in part (a), then recalculate T from the energy balance. Print out the values of T (assumed), P, nA(v), nAil), and T (recalculated). Ifthe assumed and recalculated values of T are within 0.01 K of each other, end the loop and go back to read the next set of input variables. Ifthey are not and more than 15 iterations have been performed, terminate with an error message. Otherwise, repeat the previous step, using the recalculated value of T as the assumed value for this iteration. Run the program for the eight sets of conditions shown in the spreadsheet table.