PROBLEM STATEMENT: Two fluid streams are being continuously blended in fixed proportions in a completely filled agitated reactor, as shown above. Under the conditions of operation, heat must be removed through the cooling coil at a rate Q proportional to the flow rate of the exit stream. Estimate the effect of tank diameter on permissible mass flow rates if the solution and coil-surface temperatures, T1 and T0, respectively, and the Reynolds number are to be maintained constant. Assume geometric similarity and constant fluid properties; assume further that the stream flow rates have negligible effect on the flow pattern. Neglect heat loss from the tank wall.

SOLUTION:

> restart;

The following equation [eqn 10.6-12] describes the rate, Q, at which heat is removed from the surface of the coil.

> Q := k*Int((diff(T(n),n)), A);

In order to cast the equation in dimensionless form, the function variables are rewritten in terms of the dimensionless parameters A* = A/D^2, n* = n/D, and T* = (T - T0) / (T1 - T0):

> transf:={A=Astar*D^2, n= nstar*D, T= Tstar(nstar)*(T1-T0) + T0};

These dimensionless variables are then utilized in Maple through the use of PDEtools:

> with(PDEtools, dchange);

The resulting dimensionless equation for heat is a function of diameter and the quantity dT*/dn*, the term (T1 - T0) being a constant as determined by the boundary conditions.

> Q:= dchange(transf, Q, [Astar, Tstar, nstar]);

The differential term may be rewritten as a function of the Reynolds and Prandtl number, as defined by the following:

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

> Pr:= Cp*mu/k;

> Q:=K*(T1-T0)*D*Int(Psi(Gr,Pr), A);

The integral is composed of constants, giving the term a constant value. With all other parameters being constant, the heat loss equation becomes a function of diameter alone:

> Q:=K*(T1-T0)*D*int(Psi(Gr,Pr), A);

Given that the heat removed through the cooling coil is proportional to the mass flow rate of the exit stream, this equation allows one to relate the permissible mass flow rate in terms of the system diameter.