Department of Energ Poltecnco d Mlano a Lambruschn 4 2056 MILANO Exercses of undamentals of Chemcal Processes Prof. Ganpero Gropp Exercse 7 ) Estmaton of the composton of the streams at the ext of an sothermal unt A unt s mantaned at a temperature of 80 C and at a pressure of 0 Pa. A mxture of ntromethane, acetone and acetontrle (molar composton reported n the table), s sent to the unt. Assumng deal gases and deal lqud mxture: ) calculate the bubble pressure and the dew pressure of the mxture at the temperature of the unt 2) calculate the bolng temperature and the dew temperature of the mxture at the pressure of the unt 3) evaluate f a vapor-lqud equlbrum condton s reached at the temperature and pressure of the unt 4) estmate the vaporaton rato and the composton of the lqud stream and of the vapor stream that are produced n the unt. or each spece, A, B and C are the parameters of Antone s equaton for the estmaton of the vapor pressure. Data Spece A B C Acetone 0.450 4.345 2756.220 228.060 Acetontrle 0.350 4.8950 343.00 250.523 Ntromethane 0.200 4.753 333.700 227.600 ln SAT B SAT P A P Pa T C C T
2) Estmaton of the temperature and the composton of the streams at the ext of an adabatc unt A unt s mantaned at 00 Pa pressure. A mxture of n-hexane, n-octane and n-decane, whose composton n terms of molar fractons ( ) s reported n the table, s sent to the unt. The mxture fed to the unt s at 30 C and 304 Pa. Assumng deal gases and deal lqud mxture, and nowng that the drum s adabatc, calculate the temperature and the composton of the streams that ext the unt. or each spece, the specfc heat n the lqud phase, the specfc heat n the vapor phase and the enthalp of vaporaton ( H AP, at the normal bolng temperature) can be assumed as constant. The enthalpes of vaporaton are gven at the bolng temperature at atm. Data Speces A B C lq gas C P C P H AP (T N bol) [J/mol/ C] [J/mol/ C] [J/mol] n-hexane 0.300 3.893 2696.040 224.37 200 82 28850 n-octane 0.350 3.9346 323.30 209.635 255 240 3440 n-decane 0.350 3.9748 3442.760 93.858 35 298 38750
Exercse ) Isothermal Gven that the pressure and the temperature of the unt are nown, 2N +2 unnown values must to be found, N beng the number of the speces n the unt. The equatons needed n the sstem are: global materal balance, N materal balances for the speces, and N equlbrum condtons for the -th spece between the lqud and the gas phase. L L x... N speces T,P,,x x... Nspeces s the molar fracton of the -th spece fed to the unt, the nlet molar flow rate, x s the molar fracton of the -th spece n the lqud stream L, s the molar fracton of the -th spece n the vapor stream, are the equlbrum constants for the materal transfer between the lqud and the vapor phase. In general, for each spece, the equlbrum constant depends on the temperature, the pressure and the composton of both the lqud and the vapor phases. In the exercse, gven that the gas s deal and the mxture s deal (.e. the Raoult s law s vald), t s possble to wrte the constants n a smple form, startng from the Antone s equaton for the estmaton of the saturaton pressures of the speces: T,P P T P SAT It s convenent to rearrange the problem b ntroducng the vaporaton rato : n ths wa, the problem of the determnaton the compostons of the lqud and the vapor phases s reduced to the determnaton of the vaporaton rato. Once ths rato s nown, the compostons are easl determned. The vaporaton rato s the fracton the feed flow that s vapored n the chamber. x B substtuton of the equlbrum condton x n, one obtans: x x x
The equaton requred to solve for the vaporaton rato s the constrant on the molar fractons x : 0 Eq. x peces peces Ths equaton (Eq. ) s non-lnear and must be solved wth a proper numerc procedure. In ths case, two solvng roots are fund for : a trval soluton s alwas found for = 0, whch corresponds to the absence of vaporaton. ollowng the same procedure of equaton, a second equaton s found (Eq. 2) for the soluton of the vaporaton rato b substtuton of the equlbrum constant relatonshp n x, as follows: 2 0 Eq. peces peces Equaton 2 s also non-lnear and must be solved numercall. Also n ths case, two solvng roots are found for the rato: = s a trval soluton for ths equaton, whch corresponds to the complete vaporaton of the ncomng feed. If the functons correspondng to equatons and 2 are consdered (g. a), t s noted that both the curves show a mnmum. Addtonall, unless an approprate value of s assgned as a frst attempt, some numercal procedures can dverge.
gure solvng functons Eq and 2 as a functon of the vaporaton rato. If equatons and 2 are combned (b subtracton), equaton 3 s found, also nown as the Rachford-Rce equaton. Ths equaton s stable from a numercal vewpont, snce t decreases monotoncall and has onl one solvng root between 0 and (g. b). peces peces x 0 Eq. 3 Before addressng the numercal soluton, t s convenent to verf whether or not the operatng condtons allow for a non-trval soluton of the vaporaton rato. In order to perform ths verfcaton, t s suffcent to calculate the bubble pressure and the dew pressure of the feed stream at the temperature and to verf f the pressure of the unt s comprsed n between: P dew T P P T bubble If ths condton s verfed, the vaporaton rato s comprsed between 0 and, and a lqud current and a vapor current exst at the equlbrum under the operatng condtons of the unt. In the exercse, the dew and bubble pressures are: P BUBBLE = 40 Pa P DEW = 03 Pa
Smlarl, the dew temperature and the bubble temperature can be calculated at the pressure and a vaporaton rato exsts between 0 and n case the temperature s comprsed between ther values: T BUBBLE = 72. C T DEW = 82.0 C In the exercse, a non-trval soluton exsts. The fnal vaporaton rato has the followng value: = 0.807 Wth ths value, the followng compostons are calculated for the outcomng streams: Speces P SAT [Pa] K x Acetone 24.32.948 0.496 0.254 Acetontrle 96.43 0.876 0.34 0.389 Ntromethane 50.402 0.458 0.63 0.357 Exercse 2) Adabatc lash Unt In ths case, the pressure of the unt s nown, but the temperature s unnown. Compared to the solvng sstem of the prevous exercse, n ths case an addtonal equaton s requred, whch s the enthalp balance on the unt. Gven that the unt s adabatc, the enthalp balance reads as follows: H L ) H ) H ) 0 OUT OUT Smlarl to the prevous case, t s convenent to reduce the problem to the determnaton of two unnowns onl, the vaporaton rato and the temperature T, nstead of solvng for the compostons of the two streams. The enthalp balance must be coupled wth the Rachford-Rce equaton: peces T T, P, P 0 In the exercse, the mxtures are deal and the gas s deal. The equlbrum constants are a functon of the pressure of the unt and of ts temperature, whch s unnown. Therefore, the Rachford-Rce equaton must be solved smultaneousl wth the enthalp balance, gvng rse to a sstem of two equatons n two unnowns, T and. The enthalp balance can be wrtten as:
h h ) h ) h ) L x h ) ) 0 L x h ) 0 L rom the estmaton of the dew temperature and of the bubble temperature of the nlet feed, the nlet feed s lqud, beng ts temperature (30 C) lower than the bubble temperature: T T bubble dew Pfeed 49.3 P 90.34C feed C In the enthalp balance, t s possble to choose as the reference the nlet feed, taen n the lqud state, at the temperature T, pressure P feed and composton. Thus, the enthalp of the nlet feed s null: h ) 0 Chosen the reference, t s possble to derve the specfc molar enthalp of the speces n each of the outcomng streams. or the lqud current, neglectng the pressure effects, t results: L h LASH ) c p, T T Lq T dt or the vapor stream, for each spece, the value of the H AP n the data s gven at the normal bolng pont (that s, at the bolng temperature of the pure spece at atm, P = atm, T = T N bol). It s then reasonable to follow the enthalp pathwa: h ) N T bol, Lq cp T T AP N, dt H bol, ) TLASH ap cp, N T bol, T dt or both phases, the specfc heats are consdered constant. B substtutng the specfc molar enthalp values n the enthalp balance, the second equaton of the sstem s obtaned: Lq N AP N ap N Lq c p, T bol, T H bol, ) c p, T LASH Tbol, x c p, T LASH T 0
An teratve procedure can be adopted for the numercal soluton. rst, t s possble to solve the enthalp balance as a functon of the vaporaton rato b assumng a temperature; then, the Rachford-Rce equaton s solved for the T value, gven the value of calculated n the prevous step. The procedure s then repeated up to the convergence. Dfferentl from the sothermal, n the adabatc t s not possble to verf a pror the exstence of a non-trval soluton of the vaporaton rato. The condton: T T bubble P s n fact onl necessar, but not suffcent for the exstence of a vaporaton rato dfferent from 0 or. On the one hand, the depressuraton of the nlet feed from P to P as a consequence of the soenthalpc lamnaton ma not be suffcent to vapore part of the nlet current; on the other hand, the coolng of the nlet temperature from T to T feed ma brng the nlet feed under ts bubble temperature at P. In the present case, the followng values can be used as frst attempts: = 0.5 and T = 5 C. At the end of the teratve procedure, the solvng values are: 0.679 T 0.366C The compostons of the outlet currents are: K x T N bolng [ C] h lqud h vapor [J/mol] [J/mol] n-hexane 3.86 0.699 0.29 68.700-77.09 2837.487 n-octane 0.650 0.242 0.372 25.599-549.036 202.999 n-decane 0.43 0.058 0.409 74.097-204.008 329.648