v07.08.04 007, Michael E. Hanyak, Jr., All Rights Reserved age 7-8
Energy Balance Intrductin Functinal frm: Hw t evaluate hmix,,? [ X H 0.60* + 0.40* + Δ () mix Ĥ is mlar enthalpy (kj/gml) f mixture at 40ºC and atm. Ĥ is mlar enthalpy (kj/gml) f pure n-pentane at 40ºC and atm. Ĥ is mlar enthalpy (kj/gml) f pure n-hexane at 40ºC and atm. ΔĤ mix is mlar enthalpy f mixing that is assumed zer fr ideal slutin. H + ΔH [40 C, atm, liq,, h zer n- reference state () H + ΔH [40 C, atm, liq,, h zer n- reference state (3) Ĥ is mlar enthalpy f pure n-pentane at arbitrarily-selected reference state. Ĥ is mlar enthalpy f pure n-hexane at arbitrarily-selected reference state. ΔĤ is mlar enthalpy change t g frm n- reference state t 40ºC and atm. ΔĤ is mlar enthalpy change t g frm n- reference state t 40ºC and atm. Enthalpy Change f a ure Substance is represented by an exact differential: d i c h, d + i V V d (4) is mlar heat capacity f cmp. fr a liquid, gas, r slid in able B. f F&R, 3 rd Ed. h c, hus the mlar liquid enthalpy changes fr pure n-pentane and pure n-hexane are: ΔH Δ ( Units f kj/gml Excel AEx Add-In Functin 40 C 3 5 55.4 0 + 43.68 0 ) d enthalpy("n-pentane",pt,,"c","l") C 3 ( 6.3 0 ) 40 d enthalpy("n-hexane",pt,,"c","l") 40ºC, pt, hx pt atm, hx atm v07.08.04 007, Michael E. Hanyak, Jr., All Rights Reserved age 7-9
/* Quest 4.WM.A - Energy Balance Intrductin Example rblem */ CHEG 00, Michael Hanyak, Octber 7, 005 Mathematical Mdel /* ttal */ n + n - n3 0 /* pt pentane */ 0.40*n + 0.60*n - n3pt 0 /* hx hexane */ 0.60*n + 0.40*n - n3hx 0 /* pt ml frac */ n3pt n3*x3pt /* hx ml frac */ n3hx n3*x3hx /* energy bal'n */ n*h + n*h - n3*h3 0 /* mlar enthalpies f mixtures,, and 3 in kj/gml */ h 0.40*( Hpt + dhhat("n-entane_l",pt,) ) + 0.60*( Hhx + dhhat("n-hexane_l",hx,) ) h 0.60*( Hpt + dhhat("n-entane_l",pt,) ) + 0.40*( Hhx + dhhat("n-hexane_l",hx,) ) h3 x3pt*( Hpt + dhhat("n-entane_l",pt,3) ) + x3hx*( Hhx + dhhat("n-hexane_l",hx,3) ) /* Givens: */ n 45 // gml/h n 55 // gml/h 0 // C, inlet Stream 40 // C, inlet Stream /* Ref. emp: */ pt 0 // reference state fr pure pentane: pt, C & pt atm /* Ref. emp: */ hx 0 // reference state fr pure hexane: hx, C & hx atm /* Ref. enthalpy: */ Hpt 0 // mlar enthalpy fr pure pentane at pt, C & pt atm /* Ref. enthalpy: */ Hhx 0 // mlar enthalpy fr pure hexane at hx, C & hx atm E-Z Slve Numerical Slutin 0 C 40 C 3 30.809 C n 55 gml/h n3 00 gml/h n3hx 49 gml/h n3pt 5 gml/h x3hx 0.49 ml frac x3pt 0.5 ml frac h 0 kj/gml h 3.754 kj/gml h3.0638 kj/gml v07.08.04 007, Michael E. Hanyak, Jr., All Rights Reserved age 7-0
Energy Balance Intrductin Why Arbitrary Cmpnent Reference States? ( n reactins ) Begin with the energy balance t mix tw prcess streams t frm a third ne. hese three streams labeled,, and 3 are mixtures f tw chemical cmpnents n-pentane () and n-hexane (). When we determine the mlar enthalpy f a mixture ( H i, kj/kgml), we can arbitrarily pick a reference state fr each cmpnent ( & ) and assign zer t the mlar enthalpy f the pure cmpnent at that reference state ( H 0 ). Why can we d this? Let s lk at the fllwing derivatin fr the answer. Eqn # Derivatin Cmments n + n n 0 start with the energy balance 3 3 H hmix[, X H x, H + x, H + H mix Multiplying Eq. with n, gives: n n + n ideal slutin n hmix[, n, ' s 3,, Mlar enthalpy f pure n-pentane in Eq. 3 is calculated by: 4 H [,, h + [,, h,, h units f kj/kgml term is set arbitrarily term is the enthalpy change t g frm t zer fr pure n- pure n- reference state t mixture state Simplifying the ntatin in Eq. 4 gives: 5 H H + [ Similarly, we can find the mlar enthalpy f pure n-hexane by: 6 H H + [,, mlar enthalpy fr pure n- mlar enthalpy fr pure n- Substituting Eqs. 5 and 6 int Eq. 3 gives: 7 n H n H + [, { } { + n H + [ } n hmix[, n, ' s kj / h,,, Applying the same prcedure used in Eqs. thru 7, the energy cntents f Streams and 3 are: 8 n H n, H + H [ 9 n H n H + [, { } { + n, H + H [ } { } { + n H + [ } n hmix[, n, ' s kj / h, n 3 3 hmix[ 3, 3, n 3, ' s kj / h 3 3 3, 3 3 3, 3, Ĥ in Eqs. 7, 8, and 9 is at the reference state f Ĥ in Eqs. 7, 8, and 9 is at the reference state f and fr n-pentane. and fr n-hexane. 3 Substituting Eqs. 7, 8, and 9 int the energy balance f Eq. and factring the reference enthalpy terms gives: cmpnent reference states 0 { n, + n, n3, } H + { n, + n, n3, } H + n [ +, H n, H [ + n [ +, H n, H [ n [, n [, 0 3, 3 3 3, 3 3 energy balance equatin v07.08.04 007, Michael E. Hanyak, Jr., All Rights Reserved age 7-
Energy Balance Intrductin Arbitrary Cmpnent Reference States, Cnt d ( n reactins ) Eqn # Derivatin Cmments Remember that the tw cmpnent material balances are: n, + n, n 3, fr n-pentane n, + n, n 3, fr n-hexane 0 0 Substituting Eqs. and int the energy balance f Eq. 0 gives: zer + zer Each Ĥ is multiplied by 0 n [ +, H n, H [ + n [ +, H n, H [ n [ 3, H 3, 3 n 3, H [ 3, 3 0 energy balance equatin 3 { } { } + Cnclusins In the energy balance f Eq. 3, each cmpnent reference enthalpy f Ĥ is multiplied by its cmpnent balance. hus an Ĥ is always multiplied by zer. Because f this fact, we can arbitrarily pick a reference temperature ( ) and pressure ( ) fr a cmpnent and assign any value t its reference enthalpy ( Ĥ ). Fr cnvenience, we usually assign zer t a cmpnent reference enthalpy, H 0. he reference temperature ( ) and pressure ( ) fr each cmpnent des nt have t be the same; they all culd be different. Hwever, nce yu pick a reference state fr a cmpnent, yu must use that cmpnent reference state in any enthalpy mixture (hmix) calculatins. In the energy balance, the enthalpy changes ( ' s ) are calculated frm the reference state f the cmpnents t the state f the mixtures. In Eq. 3, they are expressed fully as fllws: Frm. H [, h,, h. H [, h,, h. H [, h,, h. H [, h,, h 3. H [ 3, 3, h3,, h 3. H [,, h,, h units f kj/kgml units f kj/kgml units f kj/kgml units f kj/kgml units f kj/kgml 3 3 3 units f kj/kgml Each f these functinal frms fr a pure cmpnent can be represented by either a graph, a table, a set f equatins, r a cmputer prgram. v07.08.04 007, Michael E. Hanyak, Jr., All Rights Reserved age 7-