(%i1) | kill(all)$ |
part 2 : intro staged-process models ( problem model )
[email protected],31/01/2017
problem :(n+1)= log(phi)/log(theta)
theta = L/(k*V)
1e)if log(theta)= 0 and log(phi)= value : (n+1) → infinity
2e)if log(theta)= 0 and log(phi)= 0 : (n+1) → indeterminate
theta = 1 = L/(k*V)
(%i5) |
eq1:phi1 = -(y[i+1]-x[0]*k)*theta-y[i+1]+y[1]; eq2:phi2 = x[0]*k-y[1]; eq3:phi = rhs(eq1)/rhs(eq2); eq4:theta = L/(k*V); eq: m = n+1; |
indeterminate ⇒ L'Hopital rule = 0/0 = phi'/theta'
(%i8) |
eq5:diff(rhs(eq3),theta,1); eq6:diff(lhs(eq4),theta,1); eq7:rhs(eq) = eq5/eq6; |
cascade: plate : distillation operation
1e) operating line : y[i+1]=a*x[i]+b
2e) relative volatility : alpha = y[i]/x[i]/((1-y[i])/(1-x[i]))
1e) operating line :
(%i10) |
eq8:a = L/V; eq9:b = y[1]-x[0]*L/V; |
(%i11) | eq10:y[i+1]=rhs(eq8)*x[i]+rhs(eq9); |
2e) relative volatility = alpha :
(%i12) | eq11:alpha = y[i]/x[i]/((1-y[i])/(1-x[i])); |
(%i13) | eq12:solve(eq11,y[i])[1]; |
(%i14) | eq13:subst(i+1,i,eq12); |
(%i15) | eq14:rhs(eq10)=rhs(eq13); |
(%i16) | eq15:lhs(eq14)-rhs(eq14)=0; |
total condenser : y[1]=x[0]
(%i17) | eq16:subst(x[0],y[1],eq15); |
(%i18) | eq17:radcan(-(alpha*x[i+1])/((alpha-1)*x[i+1]+1)+(L*x[i])/V-(x[0]*L)/V+x[0]); |
(%i20) |
eq18:factor(part(eq17,1)); eq19:factor(part(eq17,2)); |
(%i23) |
eq20:coeff(eq18,x[i]); eq21:coeff(eq18,x[i+1]); eq22:ev(eq18,x[i]=0,x[i+1]=0); |
(%i26) |
eq23:factor(ratsubst(t,x[i]*x[i+1],eq18)); eq24:coeff(eq23,t); eq25:t=x[i+1]*x[i]; |
Ricatti : equation
part 3: How to solve Ricatti equation
x[i+1]*x[i]+A*x[i+1]+B*x[i]+C = 0
(%i30) |
eq26: A = eq21/eq24; eq27: B = eq20/eq24; eq28: C = eq22/eq24; |