p476.html

Example 15.5-2 Operation of a simple Temperature Controller

> restart;

> eqliq:=diff(Ul(t),t)=Q(t)+wCpl*(Ti-T(t));

> eqcoil:=diff(Uc(t),t)=-Q(t)+Qc(t);

> Ul:=t->rVCl*(T(t)-Tbar);

> Uc:=t->rVCc*(Tc(t)-Tbar);

> Q:=t->UA*(Tc(t)-T(t));

> Qc:=t->b*(Tmax-T(t));

> eqliq; eq. 15.5-15

> eqcoil; eq. 15.5-16

> eqss:=rhs(eqliq)+rhs(eqcoil); Note that this does not include Tc(t)

> Tss:=solve(eqss,T(t)); The steady state solution for any value of Ti.

> Tinit:=subs(Ti=T0,Tss); eq. 15.5-17. This is what the book calls T(0).

> Tinf:=subs(Ti=T1,Tss); eq. 15.5-19. This is what the book calls T(infinity).

> Tcss:=solve(rhs(eqcoil),Tc(t)); The steady state for the coil for any given T(t)

> Tcinit:=simplify(subs(T(t)=Tinit,Tcss)); eq. 15.5-18, but note that T(0) which we called Tinit has not been substituted into eq. 15.5-18

> Tcinf:=simplify(subs(T(t)=Tinf,Tcss)); eq. 15.5-20 and Tinf has not been substituted into 15.5-20.

> Ti:=T1;

> tr:={T=Tinf+Theta*(T0-Tinf),Tc=Tcinf+Thetac*(Tc0-Tcinf),t=rVCl*tau/UA};

> with(PDEtools[dchange]);

> lst:={eqliq,eqcoil}; Putting the two DEs in a list. This turns out not to be useful, but does demonstrate that dchange can handle multiple equations in a single call to it.

> rVCc:=rVCl/R; wCpl:=UA*F; b:=B*UA; eq. 15.5-24. R is the ratio of the heat capacity of the tank to the coil. eq. 15.5-25. F is the ratio of the heat transfer coef to the flow heat capacity. eq. 15.5-26. B is the ratio of the control action to the heat transfer coef.

> newlst:=simplify(dchange(tr,lst,[Theta(tau),tau,Thetac(tau)]));

> eqtheta:=simplify(newlst[1]*(F+B)/UA/(T0*(F+B)-F*T1-B*Tmax)); Here I had to look at each equation separately. Note that Maple may oder the two equations in newlst differently on different occasions.