example.html

Example 15.4-2 in BS&L :

For a similar version of this example click here.

Parallel Flow :

> restart;

> eqh:=w*Cph*D(Th)(z)=U0*2*Pi*r0*(Tc(z)-Th(z)); Balance on the hot side using eq. 15.4-10 and eq. 13.1-8.

> eqc:=w*Cpc*D(Tc)(z)=U0*2*Pi*r0*(Th(z)-Tc(z)); Balance on the cold side.

> s:=dsolve({eqh,eqc,Tc(0)=Tc1,Th(0)=Th1},{Tc(z),Th(z)}):

> assign(s); Th:=unapply(Th(z),z);Tc:=unapply(Tc(z),z);

[Maple Math]

[Maple Math]

Temperature changes for parallel flow:

> dT1:=simplify(Th(0)-Tc(0));

> dT2:=simplify(Th(L)-Tc(L));

[Maple Math]

> log(dT1/dT2);

[Maple Math]

> logdT12:=simplify(%,assume=positive); Agrees with eq. 15.4-14

> Cpc:=-Qh/(wc*(Tc2-Tc1)); Using eq. 15.4-9 and the fact that Qc=-Qh.

> Cph:=Qh/(wh*(Th2-Th1)); Using eq. 15.4-8.

> logdT12;

[Maple Math]

> Qc:=solve(-logdT12=logdT21,Qh); eq. 15.4-15

In this last expression, logdT21 is log(dT2/dT1) which is -log(dT1/dT2) or log((Th2-Tc2)/(Th1-Tc1)). Therefore, for parallel flow, Qc is:

> Qc:=(UA,Tc2,Tc1,Th2,Th1)->UA*((Th2-Tc2)-(Th1-Tc1))/log((Th2-Tc2)/(Th1-Tc1));

[Maple Math]

Counter Flow :

> restart;

> eqh:=w*Cph*D(Th)(z)=U0*2*Pi*r0*(Tc(z)-Th(z)); Balance on the hot side using eq. 15.4-10 and eq. 13.1-8.

> eqc:=w*Cpc*D(Tc)(z)=U0*2*Pi*r0*(Th(z)-Tc(z)); Balance on the cold side.

> s:=dsolve({eqh,eqc,Tc(0)=Tc1,Th(0)=Th1},{Tc(z),Th(z)}):

> assign(s); Th:=unapply(Th(z),z);Tc:=unapply(Tc(z),z);

[Maple Math]

[Maple Math]

Temperature changes for counter flow:

> dT1:=simplify(Th(0)-Tc(L));

[Maple Math]

> dT2:=simplify(Th(L)-Tc(0));

[Maple Math]

> log(dT1/dT2);

[Maple Math]

> logdT12:=simplify(%,assume=positive);

[Maple Math]

> Cpc:=-Qh/(wc*(Tc2-Tc1)); Using eq. 15.4-9 and the fact that Qc=-Qh.

> Cph:=Qh/(wh*(Th2-Th1)); Using eq. 15.4-8.

> logdT12;

[Maple Math]

> Qc:=solve(-logdT12=logdT21,Qh);

[Maple Math]

Therefore, for the different temperature changes for counter flow, Qc is:

> Qc:=(UA,Tc2,Tc1,Th2,Th1)->UA*((Th1-Tc2)-(Th2-Tc1))/log((Th1-Tc2)/(Th2-Tc1));

[Maple Math]