{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Helvetica" 1 9 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 10 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 97 116 101 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 152 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 238 63 192 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Map le Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 " " 0 1 234 135 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helveti ca" 0 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 3 " -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 5" -1 260 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 6" -1 261 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 7 " -1 262 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 8" -1 263 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 25 "Binary Batch Distillation " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "We will begin by creating a list of component names" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "components:=[Benzene, Toluen e];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+componentsG7$%(BenzeneG%(Tol ueneG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "In order to shorten the equations that we shall derive we will identify the components using \+ numerical subscripts that represent the position of the component in t he list of their names:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "label := proc(i)\n global components;\n local pos;\n member(i,comp onents,`pos`);\n RETURN(pos);\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "Other methods of labelling the components are possible (first \+ letters, chemical formulae etc); the method used here happens to be si mple and widely used. The list of component identities now becomes" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "compid := map(label,compone nts);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'compidG7$\"\"\"\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The number of components is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "nc := nops(components);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ncG\"\"#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 91 "Now for the equations that model the process. The amoun t of liquid in the still is given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "DiffEqns :=Diff(L,x[2])=L/x[2]/(K[2]-1): DiffEqns;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$%\"LG&%\"xG6#\"\"#*(F'\"\" \"F(!\"\",&&%\"KGF*F-F.F-F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "T he composition of the vapor leaving and the liquid remaining in the st ill is related by the equilibrium equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "i:='i': Eqmeqn := i->y[i]=K[i]*x[i]: Eqmeqn(i);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"yG6#%\"iG*&&%\"KGF&\"\"\"&%\"xGF &F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 1 "K" } {TEXT -1 107 "'s are the so-called K-values or equilibrium ratios, the calculation of which will be discussed below. The " }{TEXT 260 1 "K" }{TEXT -1 77 "'s are, in general, complicated functions of the composi tion of both phases (" }{XPPEDIT 18 0 "x[j],j=1..c" "6$&%\"xG6#%\"jG/F &;\"\"\"%\"cG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[j],j=1..c" "6$&% \"yG6#%\"jG/F&;\"\"\"%\"cG" }{TEXT -1 16 "), temperature (" }{TEXT 257 1 "T" }{TEXT -1 16 ") and pressure (" }{TEXT 258 1 "P" }{TEXT -1 223 "); the latter variables not appearing explicitly in any of the ab ove expressions. We have a binary system so we need two of these equat ions. The system of equations is completed by forcing the mole fractio ns to sum to unity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "i:='i ': AEqns := [seq(Eqmeqn(i),i=compid),add(y[i],i=compid)=1,add(x[i],i=c ompid)=1]: AEqns;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/&%\"yG6#\"\"\" *&&%\"KGF'F(&%\"xGF'F(/&F&6#\"\"#*&&F+F0F(&F-F0F(/,&F%F(F/F(F(/,&F,F(F 4F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The K-values for this sy stem are to be computed using Raoult's law" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 42 "RAOULT := i->K[i] = P[i,sat]/P: RAOULT(i);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"KG6#%\"iG*&&%\"PG6$F'%$satG\"\"\" F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 256 1 "P " }{TEXT -1 25 " is the system pressure, " }{XPPEDIT 18 0 "P[i,sat]" " &%\"PG6$%\"iG%$satG" }{TEXT -1 89 " is the vapor pressure. We form a l ist of the K-values for all species to be used later. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Kvalues := [seq(RAOULT(i),i=compid) ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(KvaluesG7$/&%\"KG6#\"\"\"*&&% \"PG6$F*%$satGF*F-!\"\"/&F(6#\"\"#*&&F-6$F4F/F*F-F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Vapor pressures can be computed using the Antoi ne equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Antoine := \+ (T, A, B, C) -> 10^(A - B / (T + C));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(AntoineG:6&%\"TG%\"AG%\"BG%\"CG6\"6$%)operatorG%&arrowGF+)\"# 5,&9%\"\"\"*&9&F3,&9$F39'F3!\"\"F9F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Parameters for this equation are as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "params:=\{AntC[Benzene] = 220.79, \+ AntB[Benzene] = 1211.033, AntA[Benzene] = 6.90565, AntC[Toluene] = 219 .482, AntB[Toluene] = 1344.8, AntA[Toluene] = 6.95464\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "which we modify below to use the indexing system employed here." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A ntparams:=subs(seq(components[i]=i,i=compid),params);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*AntparamsG<(/&%%AntCG6#\"\"\"$\"&z?#!\"#/&F(6# \"\"#$\"'#[>#!\"$/&%%AntBGF0$\"&[M\"!\"\"/&%%AntAGF0$\"'kap!\"&/&F7F)$ \"(L5@\"F4/&F=F)$\"'l0pF@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "In \+ order to carry out the numerical compuations we substitute the paramet ers into the K-value model equations. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Kval2 := subs(seq(P[i,sat]=Antoine(T,AntA[i],AntB[i], AntC[i]),i=compid),Antparams,Kvalues): Kval2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"KG6#\"\"\"*&)\"#5,&$\"'l0p!\"&F(*$,&%\"TGF($\"&z ?#!\"#F(!\"\"$!(L5@\"!\"$F(%\"PGF6/&F&6#\"\"#*&)F+,&$\"'kapF/F(*$,&F2F ($\"'#[>#F9F(F6$!&[M\"F6F(F:F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "and the complete set of (differential-algebraic) equations formed by combining the differential and algrbraic equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "DAEqns := [DiffEqns,op(AEqns)]: DAE qns;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/-%%DiffG6$%\"LG&%\"xG6#\"\" #*(F(\"\"\"F)!\"\",&&%\"KGF+F.F/F.F//&%\"yG6#F.*&&F2F6F.&F*F6F./&F5F+* &F1F.F)F./,&F4F.F;F.F./,&F9F.F)F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "into which we substitute the K-value model" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "DAEqns2:=subs(Kval2,DAEqns): DAEqns2;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7'/-%%DiffG6$%\"LG&%\"xG6#\"\"#*(F(\" \"\"F)!\"\",&*&)\"#5,&$\"'kap!\"&F.*$,&%\"TGF.$\"'#[>#!\"$F.F/$!&[M\"F /F.%\"PGF/F.F/F.F//&%\"yG6#F.*()F3,&$\"'l0pF7F.*$,&F:F.$\"&z?#!\"#F.F/ $!(L5@\"F=F.F@F/&F*FDF./&FCF+*(F2F.F@F/F)F./,&FBF.FSF.F./,&FQF.F)F.F. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The variables appearing in th ese equations are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Vars : = indets(DAEqns2,name);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VarsG<)& %\"yG6#\"\"\"&%\"xGF(&F'6#\"\"#&F+F-%\"LG%\"PG%\"TG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Note that this system of DAEs includes both te mperature and pressure as implicit variables. The number of degrees of freedom is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "nops(Vars)-n ops(DAEqns);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "In fact, there is only one degree of free dom since " }{XPPEDIT 18 0 "x[2]" "&%\"xG6#\"\"#" }{TEXT -1 149 " must be specified at the start of the integration. We specify the pressure as 1.2 atmosphere (to be converted to mm Hg) and recreate the DAE sys tem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Pspec := P = 1.2*76 0: Pspec;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"PG$\"%?\"*!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "DAEqns3 := subs(Pspec,DAEqns 2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(DAEqns3G7'/-%%DiffG6$%\"LG&% \"xG6#\"\"#*(F*\"\"\"F+!\"\",&)\"#5,&$\"'kap!\"&F0*$,&%\"TGF0$\"'#[>#! \"$F0F1$!&[M\"F1$\"+G7\\'4\"!#7F1F0F1/&%\"yG6#F0,$*&)F4,&$\"'l0pF8F0*$ ,&F;F0$\"&z?#!\"#F0F1$!(L5@\"F>F0&F,FGF0FA/&FFF-,$*&F3F0F+F0FA/,&FEF0F WF0F0/,&FUF0F+F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "The initia l value of all the algebriac variables is determined by solving the on ly algebraic equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " AEqns2:=subs(seq(Kval2[i],i=compid),AEqns): AEqns2;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7&/&%\"yG6#\"\"\"*()\"#5,&$\"'l0p!\"&F(*$,&%\"TGF($\" &z?#!\"#F(!\"\"$!(L5@\"!\"$F(%\"PGF6&%\"xGF'F(/&F&6#\"\"#*()F+,&$\"'ka pF/F(*$,&F2F($\"'#[>#F9F(F6$!&[M\"F6F(F:F6&FF(F(/,&F;F(F LF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The pressure and initial mole fractions in the liquid are fixed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "AEqns3:=subs(Pspec,x[2]=0.4,AEqns2): AEqns3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&/&%\"yG6#\"\"\",$*&)\"#5,&$\"'l0p!\" &F(*$,&%\"TGF($\"&z?#!\"#F(!\"\"$!(L5@\"!\"$F(&%\"xGF'F($\"+G7\\'4\"!# 7/&F&6#\"\"#,$)F,,&$\"'kapF0F(*$,&F3F($\"'#[>#F:F(F7$!&[M\"F7$\"+7\\'f Q%!#8/,&F%F(FAF(F(/,&F;F($\"\"%F7F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The unknown variables are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(AEqns3,name);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&&%\"yG6#\"\"\"&%\"xGF&&F%6#\"\"#%\"TG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "We solve these equations using Newton's method." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read `c:/maple/numerics/newt on.mpl`:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "start:=Newton(AEqns3,[x[1]=0 .3,y[1]=0.5,y[2]=0.5,T=80]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sta rtG7&/&%\"xG6#\"\"\"$\"+++++g!#5/&%\"yGF)$\"+x9')pyF-/&F06#\"\"#$\"+B& Q,8#F-/%\"TG$\"+@(3&e&*!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Th e complete starting point is given by adding the initial values of " } {XPPEDIT 18 0 "x[2]" "&%\"xG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "L " "I\"LG6\"" }{TEXT -1 13 " to this list" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "start2:=[x[2]=0.4,L=100,op(start)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'start2G7(/&%\"xG6#\"\"#$\"\"%!\"\"/%\"LG\"$+ \"/&F(6#\"\"\"$\"+++++g!#5/&%\"yGF3$\"+x9')pyF7/&F:F)$\"+B&Q,8#F7/%\"T G$\"+@(3&e&*!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 190 "Maple out of the box is not capable of solving DAE problems. However, we have impl emented a numerical method for solving DAE systems called BESIRK, the \+ code for which is read into Maple now." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "read `c:/maple/numerics/integ/besirk`:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "The equations are to be integrated until the liquid contains 80% toluene. The integration range is, therefore: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "xrange := 0.4..0.8;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xrangeG;$\"\"%!\"\"$\"\")F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "We integrate the DAE system using BESIRK in the next command. Note that BESIRK can take a number of opt ional commands to limit the step size, or to increase the accuracy of \+ the calculations. None of these options are needed here." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "result:=BESIRK(DAEqns3,start2,xrang e);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'resultG-%&ARRAYG6$7#;\"\"!\" #:72/6#F*7($\"1+++++++S!#;$\"$+\"F*$\"1+++++++gF2$\"1+++x9')pyF2$\"1++ +B&Q,8#F2$\"1+++@(3&e&*!#9/6#\"\"\"7($\"1+++++++XF2$\"1J>oq#)fBxF=$\"1 fHb'*fI&\\&F2$\"1=55Yk;%\\(F2$\"1$)*)*QbLe]#F2$\"1')*R$G0%*)p*F=/6#\" \"#7($\"1++++++]\\F2$\"1**4RRs*p>'F=$\"1pw*oRvd/&F2$\"1ZnkQ:cJrF2$\"1` KNh%Q%oGF2$\"1syAB'H*G)*F=/6#\"\"$7($\"1++++++b`F2$\"1Gc+\"[/l6&F=$\"1 $)y?dy>TYF2$\"14q^>&*G!y'F2$\"1#*H[![5(>KF2$\"1U3uEZ7]**F=/6#\"\"%7($ \"1+++++]>dF2$\"102f$zE)>VF=$\"1_q[r!yqF%F2$\"11+L>H*>W'F2$\"1%**p132! eNF2$\"1_Ulz,G15!#8/6#\"\"&7($\"1*********\\v/'F2$\"1WfO<+J8PF=$\"1%QQ VEq$\\RF2$\"1\"fchK*)z6'F2$\"17M%Qn5?)QF2$\"1aC$)=\"Gn,\"Fhp/6#\"\"'7( $\"1********\\zUjF2$\"1cx;nVdRKF=$\"1FX!zBLWl$F2$\"1RCmDwF4eF2$\"1lvLu Bs!>%F2$\"1#)y;4=RE5Fhp/6#\"\"(7($\"1)******\\:&3mF2$\"1g)R\"RJthGF=$ \"13T69**)*)Q$F2$\"1&ozz\"Rb;bF2$\"1?.-#3YM[%F2$\"1txTT$4`.\"Fhp/6#\" \")7($\"1)*****\\RmZoF2$\"1-8ivz6bDF=$\"1*o-F#44]JF2$\"16'[EzM-C&F2$\" 1(Q^t?l(fZF2$\"1Q-$\\`?N/\"Fhp/6#\"\"*7($\"1)****\\b(*G1(F2$\"14@36ln- BF=$\"1ECVI=3NHF2$\"1'RENn&[!)\\F2$\"1-OZEV^>]F2$\"1U-\")yh1^5Fhp/6#\" #57($\"1****\\*z2mD(F2$\"1b0U[UG#4#F=$\"1b\"*QZOdTFF2$\"1J[\"48_st%F2$ \"1p^3pyui_F2$\"1b!=mO()z0\"Fhp/6#\"#67($\"1***\\&>q%4V(F2$\"1]m$)*=3^ \">F=$\"1,C!zQ=nc#F2$\"1(Q:B7n$4XF2$\"16YoxGj!\\&F2$\"1*zd-h]V1\"Fhp/6 #\"#77($\"1++gRiG0wF2$\"14a!H0Q$[_\\<+y5Fhp/6#\"#97($\"1 +X]4_lpzF2$\"1ShO=49H9F=$\"1_PnfWsG?F2$\"1#4,*zU0cPF2$\"12*)4?d%RC'F2$ \"1$z7Q34Y3\"Fhp/6#F+7($\"1+++++++!)F2$\"1Yx!f>gTS\"F=$\"1*\\ve8c%**>F 2$\"1d\"\\'pPm7PF2$\"1Q3NIiL(G'F2$\"1on`: " 0 "" {MPLTEXT 1 0 30 "l1:=result[Lastpoint(result)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#l1G7($\"1+++++++!)!#;$\"1Yx!f>gTS\"!#9$\"1*\\ve8c%** >F($\"1d\"\\'pPm7PF($\"1Q3NIiL(G'F($\"1on`: " 0 "" {MPLTEXT 1 0 46 "[seq(lhs(start2[i])=l1[i],i=1..nops(start2))];" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7(/&%\"xG6#\"\"#$\"1+++++++!)!#;/%\"LG $\"1Yx!f>gTS\"!#9/&F&6#\"\"\"$\"1*\\ve8c%**>F+/&%\"yGF3$\"1d\"\\'pPm7P F+/&F9F'$\"1Q3NIiL(G'F+/%\"TG$\"1on`: " 0 "" {MPLTEXT 1 0 29 "plot (\{makelist(result,1,2)\});" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG 6$727$$\"1+++++++S!#;$\"$+\"\"\"!7$$\"1+++++++XF*$\"1J>oq#)fBx!#97$$\" 1++++++]\\F*$\"1**4RRs*p>'F37$$\"1++++++b`F*$\"1Gc+\"[/l6&F37$$\"1++++ +]>dF*$\"102f$zE)>VF37$$\"1*********\\v/'F*$\"1WfO<+J8PF37$$\"1******* *\\zUjF*$\"1cx;nVdRKF37$$\"1)******\\:&3mF*$\"1g)R\"RJthGF37$$\"1)**** *\\RmZoF*$\"1-8ivz6bDF37$$\"1)****\\b(*G1(F*$\"14@36ln-BF37$$\"1****\\ *z2mD(F*$\"1b0U[UG#4#F37$$\"1***\\&>q%4V(F*$\"1]m$)*=3^\">F37$$\"1++gR iG0wF*$\"14a!H0Q$[gTS\"F3-%'COLOURG6&%$RGBG$\"#5 !\"\"F-F-" 2 173 172 172 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 -30570 8450 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "and the temperature as a function of " }{XPPEDIT 18 0 "x[2]" "&%\"xG6 #\"\"#" }{TEXT -1 3 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(\{makelist(result,1,6)\});" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'CU RVESG6$727$$\"1+++++++S!#;$\"1+++@(3&e&*!#97$$\"1+++++++XF*$\"1')*R$G0 %*)p*F-7$$\"1++++++]\\F*$\"1syAB'H*G)*F-7$$\"1++++++b`F*$\"1U3uEZ7]**F -7$$\"1+++++]>dF*$\"1_Ulz,G15!#87$$\"1*********\\v/'F*$\"1aC$)=\"Gn,\" FB7$$\"1********\\zUjF*$\"1#)y;4=RE5FB7$$\"1)******\\:&3mF*$\"1txTT$4` .\"FB7$$\"1)*****\\RmZoF*$\"1Q-$\\`?N/\"FB7$$\"1)****\\b(*G1(F*$\"1U- \")yh1^5FB7$$\"1****\\*z2mD(F*$\"1b!=mO()z0\"FB7$$\"1***\\&>q%4V(F*$\" 1*zd-h]V1\"FB7$$\"1++gRiG0wF*$\"1.Yb?5!32\"FB7$$\"1+]l\"Qfqz(F*$\"1j>_ \\<+y5FB7$$\"1+X]4_lpzF*$\"1$z7Q34Y3\"FB7$$\"1+++++++!)F*$\"1on`: \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }