Department of Energy Poltecnco d Mlano Va Lambruschn 56 MILAN Exercses of Fundamentals of Chemcal Processes Prof. Ganpero Gropp Exercse utlet temperature of a WGS reactor (Stage I for the converson of C, appled for the abatement of C to a fxed value. he gas stream extng the second stage of a eformer suppled wth ar s sent to a two-stage reactor for the converson of C, where the followng reacton occurs (Water Gas Shft, WGS: C + H C + H Wth reference to the frst stage of the converson of C, t can be assumed that the outlet gases reach the thermodynamc equlbrum composton at the ext temperature U. Determne the ext temperature U that allows for the amount of C n the outlet gas stream to be equal to.9% (volume % wth respect to the dry current. DAA: Stream : = 6 C P = atm Stream : U =? P = atm C/Σd ry =.9% Hypothess: Ideal gas, WGS s at the chemcal equlbrum
Speces Flow ate (Nm /h % Vol. Dry Bass H 56.6 N. C.56 C 8.7 Ar.7 CH. H 7 dry 6. otale 9675
Soluton he approach wth the extent of reacton s chosen to wrte the materal balances. he bass for the materal balances can be set to mol/h of ncomng dry gas. he extent of the reacton λ s defned and the composton of the outcomng speces can be wrtten, n terms of molar flow rates. Speces Flow ate (Nm /h n n (mol/h n out (mol/h H 56.6 56.6 + N.. C.56.56 - C 8.7 8.7 + Ar.7.7 CH.. H 7 6. 6. - dry 6.. + otal 9675 6. 6. Gven the mol/h nlet dry gas bass, the amount of H s calculated as follows: 7 H 6. 6 he specfc on the amount of C n the outlet dry stream s used to calculate the extent of the reacton λ: C dry.56.9 he equaton leads to λ =.7 mol/h. nce λ s calculated, the ext composton s readly derved: Speces n out (mol/h % vol. n out dry bass % vol. n out H 68.8 6.7.5 N. 9.8.8 C..9.7 C.9 8.9.66 Ar.7..7 CH..7.9 dry. - H 8.9 -.8
he ext composton s related to the outlet temperature va the hypothess of thermodynamc equlbrum on the WGS reacton. In ths case: G G (, P G ( ln k eq It follows that: G ( ln k eq In the case of the WGS reacton, assumng the deal gas approxmaton, t s obtaned: k P P y y n n 65.9 out out C H C H C H eq out out PC PH yc yh nc nh ln k.788 eq he G (98K s calculated startng from the G F(98K of the reactants and the products: G ( 98K G F, (98K In the equaton, s the stochometrc coeffcent of the -th spece n the reacton. In the same way, the H (98K s calculated as follows: H ( 98K H F, (98K G F(98K and H F(98K are found n the thermodynamc property database. he G ( at a dfferent from 98K s determned wth the Van t Hoff and Krchoff laws, whch express the dependence of the G and the H from temperature. o (ΔG (/ ΔH( P Van t Hoff
Δ H( p Cp ( a b Cp c ( d Krchoff o derve the explct expresson of the G ( as a functon of temperature, the Krchoff law s to be ntegrated frst, startng from the reference temperature ( = 98K, as follows: ΔH ( ΔH (98K Cp ( d ΔH (98K ( a Δ b Δ c Δ d d a b c d a b c d ΔH ( ΔH ( a ( b ( c ( d ( G ( G(98 H ( d For each spece, the parameters of the specfc heat are also found n the thermodynamc property database: Spece H F(98K G F(98K [J/mol] [J/mol] a b x c x 5 d x 9 H.. 7. 9.7 -.8 7.65 + C -. -88...9.55 -.596 - C -6. -7..87 -.85.789 -.7 + H -98. -96. 9.8 7. -5.6 7.5 - C P s J/mol/K. Gven the data n the table above, the followng equatons are derved: 5
C P 6.7 9.6.8. 8 H 9665.66 6.7.68.69 5.8 8 G 6 5. 77.9 he solvng equaton s:.95ln 5.6 G ln keq.788 5. 77.9.95ln 5.6.7.7.. 6 he equaton s algebrac and mplct n the only unknown (f( =, whch can be solved wth several ether numerc (for nstance, the bsecton method or the Newton-aphson method or graphc methods. Solvng the equaton, one obtans: = 55.88 K = 6.7 C A smlar soluton could have been obtaned also by usng lnearzed expressons of the G (98K and the H (98K gven n some reference texts (for specfc temperature ranges. For the WGS reacton: ( 98 9. 7 G ( 85 7. 7 G G ( s cal/mol, K < < 6 K G ( s cal/mol, 6 K < < 5 K Usng the frst lnearzed equaton, the followng solvng equaton s obtaned: 98 9.7.788.987 In ths case, = 5K = 69 C, gven =.987 cal/mol/k. 6