Example 10.6-2

An electric heating coil of diameter D is being designed to keep a large tank of liquid above its freezing point. It is desired to predict the temperature that will be reached by the coil surface as a function of the heating rate Q and the tank surface temperature T0. This prediction is to be made on the basis of experiments with a smaller geometrically similar apparatus filled with the same liquid.

Outline a suitable experimental procedure for making the desired prediction. Temperature dependance of the physical properties other than rho may be neglected, and the entire heating coil surface may be assumed to be at the same temperature, T1.

> restart;

> Q:=-k*Int(diff(T(r),r),A); This integral is performed over the entire surface area of the coil, A. The differential is performed with the conditions of the fluid immediately adjacent to the coil surface.

> `Dimensionless Variables`:={T(r)=Tstar(rstar)*(T1-T0)+T0, r=D*rstar, A=Astar*D^2};

> Q:=subs(`Dimensionless Variables`,Q);

Maple will not simplify this futher, but we can see that the T0 in the differential goes away since it is a constant. The (T1-T0) term can be pulled out of the differential. Also, the D introduced into the bottom of the differential with the change of variables can be pulled out in the denominator, leaving:

> Q:=-k*D^2/D*(T1-T0)*Int(diff(Tstar(rstar),rstar),Astar);

The dimensionless integral in this expression can also be expressed as some function of the Grasof and Prandtl numbers, psi.

> Pr:=Cp*mu/k; Gr:=rho^2*beta*(T1-T0)*g*D^3/mu^2;

The expression becomes:

> unassign('Q'); DimEq:=Q/k/D/(T1-T0)=psi(Pr,Gr);

If we multiply both sides by Gr:

> DimEq:=DimEq*Gr;

The Prandtl number can be considered constant since we are not considering the temperature dependance of the physical properties. It can be seen that the coefficient on the right hand side is equal to the Grashof number, and can therefore be included in the function, psi. If we solve for (T1-T0) and define a new function, phi, of (D^2*Q*rho^2*beta*g/k/mu^2) the the expresion simplifies to:

> DimEq:=(T1-T0)=(T1-T0)/Gr*phi(lhs(DimEq));

This equation can be plotted with experimental measurements of T1,T0, D and the physical properties to predict behavior of larger systems. Also, if the ratio of Q to D^2 is maintained, the difference in temperatures, (T1-T0), will be inversely proportional to D^3.